Non-Temporal Environmental Factor-Driven Dissolved Oxygen Prediction via Physics-Informed Regression for Sustainable Environmental Monitoring

Abstract

Dissolved oxygen (DO) is a critical indicator for assessing marine ecological health and hypoxia risk. Most existing DO prediction studies rely on time-series forecasting models, which require continuous temporal observations and are often unreliable in practical marine monitoring scenarios due to sparse sampling, missing records, and heterogeneous measurement conditions. To address this limitation, this paper investigates the problem of non-temporal DO prediction, aiming to learn a direct nonlinear mapping between environmental drivers and DO concentration. To explicitly model nonlinear pairwise interaction effects between environmental variables, we propose a Factor-Interaction Neural Network (FINN), which decomposes DO estimation into main effects and structured pairwise interaction effects. This interaction-driven design enhances both representation capacity and interpretability compared with conventional multilayer perceptrons. Furthermore, we develop a physics-informed extension, termed PI-FINN, by incorporating oceanographic-consistent regularization priors that reflect key DO formation mechanisms, including temperature-related solubility behavior, depth-wise smoothness associated with stratification, and chlorophyll-driven biological oxygen production tendencies. To evaluate the physical plausibility of model predictions beyond standard accuracy metrics, we introduce a physics-consistency assessment protocol based on Physics Consistency Violation Rate (PCVR) and its robust variant, and further analyze their convergence stability under different driver-weight configurations. Extensive experiments on a real-world marine dataset demonstrate that FINN achieves competitive predictive accuracy compared with strong machine learning baselines (e.g., SVR, Random Forest, and XGBoost), while the proposed physics-informed design mainly improves the physical consistency, robustness, and interpretability of DO estimation under heterogeneous environmental regimes, although it does not necessarily guarantee superior RMSE or MAE performance compared with purely data-driven models. Specifically, FINN achieves an RMSE of 0.3130, an $R^2$ of 0.9831, and a PCVR of 0.4826 on a dataset composed of key environmental variables, including depth, temperature, salinity, and chlorophyll-a, collected under sparse and irregular sampling conditions. Ablation studies confirm the effectiveness of both factor-interaction modeling and physics-guided regularization components. Overall, the proposed framework further provides a reliable tool for sustainable environmental monitoring by enabling physically consistent dissolved oxygen prediction under sparse observational conditions. Such capability is critical for supporting sustainable water resource management, hypoxia risk assessment, and long-term ecological protection.

1. Introduction

Dissolved oxygen (DO) is a fundamental indicator of aquatic ecosystem health, directly governing respiration and metabolic activities of organisms and regulating biogeochemical cycling processes in marine and freshwater environments. Dissolved oxygen plays a critical role in sustainable water resource management, as oxygen depletion can lead to hypoxia, biodiversity loss, and ecosystem degradation, especially in coastal and estuarine regions [1,2]. Accurate DO prediction is therefore of great importance for water quality monitoring, ecological risk assessment, early-warning systems, and adaptive resource management.

Traditional DO simulation approaches are mainly based on mechanistic models and numerical process-based frameworks, which explicitly describe oxygen transfer, biochemical reactions, phytoplankton growth, and hydrodynamic mixing. Despite their theoretical rigor, such models typically require high-resolution temporal observations, detailed parameter calibration, and comprehensive knowledge of local hydrodynamic and biogeochemical conditions. In real-world monitoring scenarios, however, marine sampling is often sparse, irregular, and incomplete due to sensor malfunction, harsh environmental conditions, and operational cost constraints, making the required temporal continuity and model inputs unavailable in many practical cases [3,4]. Existing approaches often rely on continuous temporal observations or purely data-driven learning paradigms, which not only limit their applicability under sparse observations but may also lead to predictions that violate known physical relationships. Consequently, developing reliable DO estimation methods under incomplete observational regimes remains a challenging yet highly practical problem.

Recent studies have increasingly emphasized data-driven and hybrid modeling approaches for dissolved oxygen (DO) prediction, aiming to improve predictive accuracy in complex and heterogeneous aquatic environments. For example, advanced machine learning and deep learning frameworks, including ensemble learning, transfer learning, and hybrid neural networks, have been widely adopted [5]. In particular, the integration of multi-source environmental variables with optimization strategies has been shown to significantly enhance DO prediction performance across rivers, estuaries, and aquaculture systems [6]. However, most of these approaches are inherently developed within a time-series prediction paradigm, relying on continuous temporal observations, historical sequences, or recurrent architectures (e.g., CNN-GRU and related sequence-based models), which fundamentally limits their applicability under sparse or irregular sampling conditions [7]. Moreover, purely data-driven models often lack physical interpretability and may produce predictions that violate known physical constraints, particularly in data-scarce or heterogeneous environmental regimes [8].

To address the limitations of purely data-driven approaches, recent studies have increasingly explored physics-informed neural networks (PINNs) for modeling water systems, demonstrating their capability to incorporate governing physical laws and improve prediction consistency under limited-data conditions [9]. For instance, PINNs have been successfully applied to hydrodynamic and water quality processes, such as salinity transport and shallow water flow modeling, by embedding partial differential equations and boundary conditions into the learning framework [10,11]. Recent reviews further highlight that PINNs can enhance generalization and reduce data requirements while maintaining physical consistency, making them particularly suitable for environmental systems with incomplete observations [11]. However, despite these advantages, most existing PINN-based approaches are inherently formulated within a spatiotemporal framework, requiring temporal derivatives, initial conditions, or continuous observations to solve governing equations. This strong dependence on temporal information limits their applicability in real-world scenarios characterized by sparse, irregular, or discontinuous sampling. Additionally, challenges related to model convergence, scalability, and the requirement for well-defined physical equations further constrain their practical deployment in complex environmental systems [11].

To overcome the limitations of process-based modeling, machine learning (ML) and deep learning (DL) techniques have been increasingly applied to water quality prediction tasks, including DO forecasting and related variables such as biochemical oxygen demand, chlorophyll concentration, and nutrient loads [12,13,14,15]. Ensemble learning models, such as Random Forest (RF) [16], Extreme Gradient Boosting (XGBoost) [17], and Support Vector Regression (SVR) [18], have demonstrated strong capability in nonlinear regression with heterogeneous environmental inputs [19,20]. Meanwhile, deep neural architectures including convolutional neural networks (CNNs) [21], recurrent neural networks (RNNs) [22], and long short-term memory networks (LSTMs) [23] have been widely adopted for spatiotemporal modeling in hydrological and environmental systems [8,24]. More recently, Transformer-based models have shown promising performance for long-range dependency learning in environmental time-series forecasting [25,26].

However, despite the rapid progress of ML/DL-based water quality modeling, two critical limitations remain unresolved. First, most existing DO prediction frameworks are inherently time-dependent, relying on lagged variables, sequential inputs, or continuous monitoring records. Such requirements significantly restrict their applicability in real marine monitoring systems where observations are typically intermittent and affected by missing values [2,27]. Second, purely data-driven predictors often behave as black-box approximators and may produce physically implausible predictions that violate well-established environmental principles. For example, DO solubility generally decreases with increasing temperature, and DO profiles typically vary smoothly with depth due to stratification and mixing effects. Similarly, chlorophyll-a concentration may influence DO through photosynthetic oxygen production under productive conditions, although this relationship can vary substantially depending on respiration, eutrophication, stratification, and other ecological processes. These mechanisms imply that DO is jointly governed by multiple drivers and their coupled nonlinear interactions [28,29]. Without explicitly modeling factor interactions and enforcing physical consistency, ML/DL models may overfit observational noise, learn spurious correlations, and suffer from degraded generalization under distribution shifts, limiting their reliability in ecological monitoring.

To improve robustness and interpretability, physics-aware learning paradigms have been actively studied in recent years. Physics-informed neural networks (PINNs) and related physics-guided frameworks incorporate physical laws into the training objective or model structure, thereby reducing the effective hypothesis space and improving generalization [30,31,32]. Such approaches have achieved notable success in fluid dynamics, reactive transport, climate modeling, and hydrological prediction tasks [33,34,35]. In the water quality domain, several studies have attempted to integrate process knowledge into learning-based models to enhance reliability and physical plausibility [36,37]. Nevertheless, most existing physics-informed DO prediction studies still focus on temporal forecasting settings or assume access to high-frequency sequential data, leaving the challenging problem of non-temporal DO estimation from instantaneous environmental observations largely underexplored.

Motivated by realistic marine monitoring deployments where only snapshot measurements are available, this work investigates a non-temporal DO prediction setting where each observation is treated as an independent sample and DO concentration is estimated solely from contemporaneous environmental drivers. Under such a formulation, the model must learn physically meaningful nonlinear couplings among depth, salinity, temperature, and chlorophyll-a from limited feature dimensions, while simultaneously ensuring physical plausibility.

To address this gap, we propose a novel non-temporal dissolved oxygen prediction framework that estimates DO concentration directly from four key environmental factors, including depth, salinity, temperature, and chlorophyll-a concentration, without requiring temporal features or lagged inputs. Specifically, we develop a Factor-Interaction Neural Network (FINN), which explicitly decomposes the regression function into main-effect subnetworks and pairwise interaction subnetworks. This structured design enables FINN to capture nonlinear couplings between environmental drivers in a more interpretable manner than conventional multilayer perceptrons (MLPs) [38] and is conceptually aligned with additive and interaction modeling principles such as generalized additive models (GAM) and neural additive models (NAM) [39,40]. However, unlike conventional additive formulations that assume independent variable contributions, FINN explicitly introduces structured pairwise interaction subnetworks to model nonlinear environmental couplings. Therefore, the proposed framework emphasizes interaction-aware representation learning rather than merely increasing neural network complexity. Building upon FINN, we further introduce a physics-informed extension termed PI-FINN, which incorporates differentiable physical regularization terms derived from oceanographic principles, including temperature-dependent oxygen solubility monotonicity, depth-wise smoothness priors, and chlorophyll-related oxygen production tendencies. These constraints are implemented through automatic differentiation, enabling end-to-end optimization under physical guidance. Although related ideas such as additive modeling and feature interaction learning have been explored in prior work, the proposed FINN and PI-FINN frameworks provide a structured and interpretable neural formulation specifically designed for non-temporal environmental prediction and have not been previously introduced in this form.

Beyond model construction, we emphasize that evaluating physical plausibility is equally important for reliable environmental prediction. Existing evaluation strategies often rely solely on prediction accuracy metrics (RMSE, MAE, and $R^2$), which cannot reveal whether the learned predictor violates fundamental physical relationships. Although several studies have explored monotonic violation counting as a physical inconsistency indicator, such binary metrics are often overly sensitive to noise and may impose unrealistic global monotonic assumptions across heterogeneous environmental regimes. In this paper, we propose a novel evaluation criterion named Region-Aware Soft Physics Consistency Violation Rate (RS-PCVR). RS-PCVR introduces a continuous relaxation to measure both violation frequency and severity, and activates constraints only within empirically reliable regimes. This design provides a stable and interpretable physical plausibility assessment and further enables a convergence-stability analysis framework for physics-informed learning, offering a rigorous evaluation protocol for physically guided DO prediction. By improving the reliability and physical consistency of DO prediction, this study contributes to sustainable environmental monitoring and informed decision-making.

The main contributions of this work can be summarized as follows:

• We formulate DO estimation as a non-temporal regression task that predicts DO directly from depth, salinity, temperature, and chlorophyll-a, enabling modeling under sparse and discontinuous marine observations.
• We propose FINN, a structured architecture that disentangles main effects and pairwise interaction effects among environmental drivers, improving nonlinear representational capability and interpretability.
• We develop PI-FINN by embedding differentiable oceanographic priors into the training objective, thereby enforcing physically consistent DO responses through end-to-end optimization.
• We introduce RS-PCVR, a region-aware soft physical violation metric that provides stable physical plausibility assessment and enables convergence-stability analysis for physics-informed learning.
• Extensive experiments and ablation studies against multiple ML and DL baselines validate the effectiveness and practical value of the proposed framework.

2. Materials and Methods

2.1. Problem Formulation

To facilitate non-temporal modeling, we adopt an independence assumption as a simplifying approximation, whereby each observation is treated as an independent sample without considering temporal dependencies. Let $\mathcal{D}={\left\{({\mathbf{x}}_i,y_i)\right\}}_{i=1}^N$ denote a non-temporal marine water quality dataset, where N is the total number of samples. For the i-th observation:

$${\mathbf{x}}_i={[D_i,S_i,T_i,C_i]}^{\top }\in {\mathbb{R}}^4,y_i=D{O}_i\in \mathbb{R},$$

(1)

where $D_i$ represents seawater depth (m), $S_i$ denotes salinity (psu), $T_i$ denotes water temperature (°C), $C_i$ denotes chlorophyll-a concentration (mg · m⁻³), and $D{O}_i$ is the corresponding dissolved oxygen concentration (mg · L⁻¹).

The objective of this study is to learn a regression function $f:{\mathbb{R}}^4\to \mathbb{R}$ such that

$${\widehat{y}}_i=f\left({\mathbf{x}}_i\right)\approx y_i,$$

(2)

without exploiting any temporal dependency among samples. Each observation is treated as an independent realization under quasi-stationary environmental conditions.

2.2. Overview of the Proposed Framework

To explicitly capture the nonlinear interactions among environmental factors while maintaining physical interpretability, we propose a Factor-Interaction Neural Network (FINN) and its physics-informed extension (PI-FINN). The core idea is to decompose the dissolved oxygen prediction into main effects of individual environmental variables and pairwise interaction effects between physically coupled factors. Furthermore, physically meaningful constraints are incorporated into the learning process through automatic differentiation to ensure consistency with known oceanographic principles.

Unlike GAM and NAM, which primarily assume additive independent contributions from individual variables, FINN explicitly models structured pairwise interactions among environmental drivers. To further verify that the observed improvements originate from interaction modeling rather than increased parameterization alone, ablation experiments comparing MLP and FINN are conducted in Section 3.

2.3. Factor-Interaction Neural Network (FINN)

2.3.1. Model Decomposition

The FINN model is developed in this work as a structured neural regression framework that explicitly captures both factor-wise contributions and their interactions. The FINN model decomposes the regression function as

$$f\left(x\right)=\sum _{k\in \mathcal{F}}{f}_k\left(x_k\right)+\sum _{(i,j)\in \mathcal{P}}{f}_{ij}(x_i,x_j),$$

(3)

where $\mathcal{F}=\{D,S,T,C\}$ denotes the set of environmental factors, $\mathcal{P}$ denotes a predefined set of factor pairs, $f_k(·)$ models the main effect of factor k, $f_{ij}(·,·)$ models the interaction effect between factors i and j. This formulation is inspired by additive models and interaction learning, while leveraging neural networks for nonlinear representation learning.

2.3.2. Main-Effect Subnetworks

For each environmental variable $x_k$, a dedicated subnetwork is constructed:

$${\mathbf{h}}_k=f_k(x_k;{\mathit{\theta }}_k),f_k:\mathbb{R}\to {\mathbb{R}}^d,$$

(4)

where ${\mathit{\theta }}_k$ denotes trainable parameters, d is the latent feature dimension, $f_k(·)$ is implemented as a multilayer perceptron (MLP) with ReLU activation functions. This design allows the model to learn factor-specific nonlinear transformations independently.

2.3.3. Pairwise Interaction Subnetworks

To explicitly model coupled environmental effects, interaction subnetworks are defined for selected factor pairs:

$${\mathbf{h}}_{ij}=f_{ij}([x_i,x_j];{\mathit{\theta }}_{ij}),f_{ij}:{\mathbb{R}}^2\to {\mathbb{R}}^d,$$

(5)

where $[x_i,x_j]$ denotes vector concatenation. In this study, interactions involving temperature (e.g., $T-S$, $T-C$, $D-T$) are emphasized due to their known influence on oxygen solubility and biological activity. The selection of interaction pairs is guided by prior oceanographic knowledge, ensuring that the modeled relationships are physically meaningful. Meanwhile, the overall network architecture is carefully designed to balance expressive capacity and interpretability, particularly under limited and sparse data conditions.

2.3.4. Feature Fusion and Prediction Head

All latent representations are concatenated:

$$\mathbf{h}=[{\mathbf{h}}_D;{\mathbf{h}}_S;{\mathbf{h}}_T;{\mathbf{h}}_C;{\mathbf{h}}_{ij}],$$

(6)

and passed to a regression head:

$$\widehat{y}=g(\mathbf{h};{\mathit{\theta }}_o),$$

(7)

where $g(·)$ is an MLP producing the final dissolved oxygen prediction.

2.3.5. Supervised Learning Objective

The FINN model is trained by minimizing the mean squared error (MSE):

$${\mathcal{L}}_{\mathrm{MSE}}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}{\left({\widehat{y}}_i-y_i\right)}^2.$$

(8)

2.4. Physics-Informed FINN (PI-FINN)

2.4.1. Motivation

Purely data-driven models may violate well-established physical laws governing dissolved oxygen dynamics, particularly under sparse or noisy observations. To address this issue, we integrate physics-informed constraints into the FINN framework, following the paradigm of physics-informed neural networks (PINNs). Specifically, these constraints are formulated as soft regularization terms with conditional validity, rather than being enforced as strict physical laws, thereby providing flexibility while still guiding the model toward physically consistent solutions.

2.4.2. Temperature Monotonicity Constraint

According to gas solubility theory and Henry’s law, dissolved oxygen concentration decreases with increasing temperature:

$${\displaystyle \frac{\partial DO}{\partial T}}\le 0.$$

(9)

This constraint is enforced through the penalty:

$${\mathcal{L}}_T={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}max{\left(0,{\displaystyle \frac{\partial {\widehat{y}}_i}{\partial T_i}}\right)}^2.$$

(10)

2.4.3. Depth Smoothness Constraint

Dissolved oxygen typically exhibits smooth variation along the depth dimension in the absence of abrupt mixing events. To suppress nonphysical oscillations, a second-order smoothness regularization is applied:

$${\mathcal{L}}_D={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}{\left({\displaystyle \frac{{\partial }^2{\widehat{y}}_i}{\partial D_i^2}}\right)}^2.$$

(11)

Notably, this smoothness constraint is incorporated as a soft regularization term rather than a strict physical assumption, allowing the model to retain flexibility in scenarios where local deviations from smoothness may occur.

2.4.4. Chlorophyll-Driven Oxygen Production Constraint

Chlorophyll-a concentration serves as a proxy for phytoplankton biomass, which contributes to oxygen production via photosynthesis. Thus, a weak and region-dependent positive tendency is encouraged under productive environmental conditions, rather than assuming a globally valid monotonic relationship.

$${\displaystyle \frac{\partial DO}{\partial C}}\ge 0,$$

(12)

with the penalty:

$${\mathcal{L}}_C={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}max{\left(0,-{\displaystyle \frac{\partial {\widehat{y}}_i}{\partial C_i}}\right)}^2.$$

(13)

2.4.5. Total Physics-Informed Loss

The overall training objective of PI-FINN is

$${\mathcal{L}}_{\mathrm{PI}-\mathrm{FINN}}={\mathcal{L}}_{\mathrm{MSE}}+{\lambda }_T{\mathcal{L}}_T+{\lambda }_D{\mathcal{L}}_D+{\lambda }_C{\mathcal{L}}_C,$$

(14)

where ${\lambda }_T$, ${\lambda }_D$, ${\lambda }_C$ are non-negative hyperparameters controlling the strength of each physical constraint. All derivatives are computed via automatic differentiation, ensuring end-to-end trainability.

3. Experiments

3.1. Experimental Setup

3.1.1. Dataset Description

The experiments are conducted on a marine water quality dataset collected from Mashan National Marine Ranch in the southern waters of Rongcheng Bay, Rongcheng City, Shandong Province. The dataset was collected over the period from 2016 to 2020, with an original temporal resolution of 20 min; however, temporal information is not explicitly utilized in the proposed non-temporal modeling framework. The study area corresponds to the Mashan National Marine Ranch, covering a spatial extent of approximately 345.74 hectares with an average water depth of about 11 m. After data cleaning and preprocessing, a total of 36,674 valid samples were retained for subsequent analysis. Each sample consists of five recorded variables:

$$[t_i,D{O}_i,D_i,S_i,T_i,C_i],$$

(15)

where $t_i$ is a timestamp used solely for indexing purposes and is not involved in model training. To focus on the instantaneous environmental response of dissolved oxygen, only the four environmental factors ${\mathbf{x}}_i=[D_i,S_i,T_i,C_i]$ are used as model inputs, while the dissolved oxygen concentration $D{O}_i$ is treated as the prediction target. All samples are considered independent observations under quasi-stationary environmental conditions.

3.1.2. Data Preprocessing

To ensure numerical stability and physical interpretability, the following preprocessing steps are applied: depth (D) and salinity (S) are normalized using z-score normalization; temperature (T) is normalized linearly to preserve monotonic trends; chlorophyll-a concentration (C) is transformed using $log(1+C)$ to mitigate skewed distributions; and all preprocessing parameters are computed using the training set only and applied consistently to test sets. Since no temporal dependency is considered, the dataset is randomly partitioned as follows: 80% training set and 20% test set.

3.2. Compared Methods

To comprehensively evaluate the effectiveness of the proposed models, several baseline and advanced regression methods are compared.

3.2.1. Machine Learning Models

• Support Vector Regression with RBF kernel (SVR);
• Random Forest Regressor (RF);
• XGBoost Regressor (XGBoost).

These methods serve as non-deep-learning baselines and represent commonly used approaches in environmental modeling.

3.2.2. Deep Learning Models

• MLP: A standard multilayer perceptron with fully connected layers and ReLU activation;
• FINN: The proposed Factor-Interaction Neural Network without physical constraints;
• PI-FINN: The physics-informed FINN incorporating physical consistency regularization.

All deep models are trained using the same optimization settings to ensure fair comparison.

3.3. Evaluation Metrics

3.3.1. Standard Regression Metrics

The predictive performance of all models is evaluated using the following metrics:

(1)

Root Mean Squared Error (RMSE)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{({\widehat{y}}_i-y_i)}^2}.$$

(16)

(2)

Mean Absolute Error (MAE)

$$\mathrm{MAE}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}|{\widehat{y}}_i-y_i|.$$

(17)

(3)

Coefficient of Determination

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^N{({\widehat{y}}_i-y_i)}^2}{{\sum }_{i=1}^N{(y_i-\overline{y})}^2}},$$

(18)

where $\overline{y}$ denotes the mean ground-truth dissolved oxygen value.

3.3.2. Physics Consistency Violation Rate (PCVR)

To quantitatively assess whether model predictions comply with known physical principles, a Physics Consistency Violation Rate (PCVR) is introduced.

(1)

Temperature Monotonicity Violation

$${\mathbb{I}}_T^{\left(i\right)}=\mathbb{I}\left({\displaystyle \frac{\partial {\widehat{y}}_i}{\partial T_i}}>0\right),$$

(19)

where $\mathbb{I}\left(·\right)$ is the indicator function.

(2)

Chlorophyll Positivity Violation

$${\mathbb{I}}_C^{\left(i\right)}=\mathbb{I}\left({\displaystyle \frac{\partial {\widehat{y}}_i}{\partial C_i}}<0\right).$$

(20)

(3)

Depth Positivity Violation

$${\mathbb{I}}_D^{\left(i\right)}=\mathbb{I}\left({\displaystyle \frac{\partial {\widehat{y}}_i}{\partial D_i}}<0\right).$$

(21)

(4)

Overall PCVR

$$PCVR={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}\left({\mathbb{I}}_T^{\left(i\right)}+{\mathbb{I}}_D^{\left(i\right)}+{\mathbb{I}}_C^{\left(i\right)}\right).$$

(22)

A lower PCVR indicates better adherence to physical consistency.

3.3.3. Region-Aware Soft PCVR (RS-PCVR)

To evaluate whether the predicted dissolved oxygen (DO) concentration satisfies domain physical knowledge, we introduce the Physics Consistency Violation Rate (PCVR). RS-PCVR is formulated as an empirical evaluation metric to quantify the degree of physical consistency in model predictions, rather than enforcing strict physical constraints. Existing physics-informed evaluation metrics typically adopt a hard binary violation counting strategy, which marks a sample as violated once the local derivative sign contradicts the expected physical trend. However, such a formulation suffers from three major limitations when applied to real marine water-quality data:

• Local derivative sensitivity: gradient-based monotonicity constraints are highly sensitive to measurement noise and local function oscillations.
• Overly strict global assumption: physical monotonicity is often valid only in certain environmental regimes rather than across the entire state space.
• Lack of violation severity: hard metrics treat minor and severe violations equally, leading to unstable evaluation and poor interpretability.

Therefore, we propose an enhanced metric named Region-Aware Soft PCVR (RS-PCVR), which integrates soft violation quantification with state-dependent activation of physical constraints, enabling a more robust and interpretable assessment of physical consistency.

Let the input vector be

$$\mathbf{x}={[D,S,T,C]}^T,$$

(23)

where D, S, T, and C denote depth, salinity, temperature, and chlorophyll concentration, respectively. The regression model is defined as

$$\widehat{y}=f_{\theta }\left(\mathbf{x}\right),$$

(24)

where $\widehat{y}$ denotes the predicted DO concentration, and $f_{\theta }(·)$ is a neural network parameterized by $\theta$.

To measure physical consistency, we compute the local sensitivity (first-order derivative) of the predicted DO w.r.t. each input variable:

$${\mathbb{I}}_T^{\left(i\right)}=\mathbb{I}\left({\displaystyle \frac{\partial {\widehat{y}}_i}{\partial T_i}}>0\right),$$

(25)

In practice, these derivatives are obtained through automatic differentiation.

Based on established marine environmental knowledge, DO generally follows the monotonic relationships:

DO tends to increase with depth (or at least should not decrease sharply), due to lower temperature and reduced surface reaeration disturbance in certain stratified regions:

$${\displaystyle \frac{\partial f_{\theta }}{\partial D}}\ge 0.$$

(26)

DO tends to decrease with temperature, since oxygen solubility decreases as temperature rises:

$${\displaystyle \frac{\partial f_{\theta }}{\partial T}}\le 0.$$

(27)

DO tends to increase with chlorophyll concentration under productive conditions, since chlorophyll reflects phytoplankton activity and potential oxygen generation:

$${\displaystyle \frac{\partial f_{\theta }}{\partial C}}\ge 0.$$

(28)

However, these monotonic relationships are not strictly valid globally, due to biological respiration, seasonal mixing, and multi-factor coupling. Thus, enforcing them across the entire dataset may lead to biased evaluation and unrealistic penalization, particularly for chlorophyll-related effects, which may be altered or reversed by respiration, decomposition, eutrophication, light limitation, and water-column stratification.

Instead of using binary violation indicators, we define a soft violation intensity function. For a constraint requiring a gradient $g\le 0$, the violation intensity is

$$v^{-}\left(g\right)={\displaystyle \frac{max(g,0)}{\left|g\right|+ϵ}}$$

(29)

For a constraint requiring $g\le 0$, the violation intensity is

$$v^{-}\left(g\right)={\displaystyle \frac{max(g,0)}{\left|g\right|+ϵ}},$$

(30)

where $ϵ$ is a small constant (e.g., ${10}^{-6}$) to avoid numerical instability. This design has the following desirable properties:

• $v\left(g\right)\in [0,1]$;
• $v\left(g\right)=0$ indicates strict satisfaction of the constraint;
• $v\left(g\right)\to 1$ indicates severe violation with large opposite-sign gradient magnitude;
• small gradient sign fluctuations lead to mild penalties rather than abrupt binary switching.

Since physical monotonicity assumptions are typically valid only under certain regimes, we incorporate a region-aware activation mechanism. A state-dependent indicator function are defined as follows:

$${\mathrm{\Pi }}_{\mathrm{\Omega }}\left(\mathbf{x}\right)=\left\{\begin{array}{cc}1,\hfill & \mathbf{x}\in \mathrm{\Omega }\hfill \\ 0,\hfill & \mathbf{x}\notin \mathrm{\Omega }\hfill \end{array}\right.$$

(31)

where $\mathrm{\Omega }$ denotes the regime in which a constraint is considered reliable.

In this work, we adopt the following region constraints:

Temperature monotonicity constraint is activated only in a moderate temperature range:

$${\mathrm{\Omega }}_T=\{\mathbf{x}\mid T_{low}<T<T_{high}\}$$

(32)

Chlorophyll monotonicity constraint is activated only when chlorophyll exceeds a threshold:

$${\mathrm{\Omega }}_C=\{\mathbf{x}\mid C>C_{min}\}$$

(33)

Depth monotonicity constraint is applied globally:

$${\mathrm{\Omega }}_D={\mathbb{R}}^4$$

(34)

These region constraints reflect realistic oceanographic observations: for example, chlorophyll is informative primarily in productive conditions, and the DO–temperature monotonic trend is most reliable in moderate temperature regimes rather than extreme cases.

Given a dataset with N samples ${\left\{x_i\right\}}_{i=1}^N$, we define the Region-Aware Soft PCVR (RS-PCVR) as

$$\mathrm{RS}-\mathrm{PCVR}={\displaystyle \frac{1}{Z}}{\displaystyle \sum _{i=1}^N}\left({\mathbb{I}}_{{\mathrm{\Omega }}_D}\left({\mathbf{x}}_i\right)·v^{+}\left({\mathbf{g}}_D^i\right)+{\mathbb{I}}_{{\mathrm{\Omega }}_T}\left({\mathbf{x}}_i\right)·v^{-}\left({\mathbf{g}}_T^i\right)+{\mathbb{I}}_{{\mathrm{\Omega }}_C}\left({\mathbf{x}}_i\right)·v^{+}\left({\mathbf{g}}_C^i\right)\right),$$

(35)

where

$$g_D^i={\displaystyle \frac{\partial f_{\theta }\left({\mathbf{x}}_i\right)}{\partial D}},g_T^i={\displaystyle \frac{\partial f_{\theta }\left({\mathbf{x}}_i\right)}{\partial T}},g_C^i={\displaystyle \frac{\partial f_{\theta }\left({\mathbf{x}}_i\right)}{\partial C}}$$

(36)

and Z is the total number of activated constraints:

$$Z={\displaystyle \sum _{i=1}^N}\left({\mathbb{I}}_{{\mathrm{\Omega }}_D}\left({\mathbf{x}}_i\right)+{\mathbb{I}}_{{\mathrm{\Omega }}_T}\left({\mathbf{x}}_i\right)+{\mathbb{I}}_{{\mathrm{\Omega }}_C}\left({\mathbf{x}}_i\right)\right)$$

(37)

Thus, RS-PCVR is normalized and remains comparable across datasets and experiments. The final RS-PCVR is a continuous score within $[0,1]$, where smaller values indicate stronger physical consistency.

Compared with traditional PCVR, our RS-PCVR introduces two improvements:

• Softness: violations are weighted by severity, not merely sign disagreement.
• Region-awareness: constraints are only applied when the physical assumption is considered reliable.

This yields a more stable, interpretable, and realistic metric for physics-informed evaluation in marine water-quality modeling.

In implementation, RS-PCVR is computed through PyTorch 2.6.0 automatic differentiation. For each test sample, we calculate the gradient vector ${\nabla }_x{f}_{\theta }\left(\mathbf{x}\right)$ and evaluate soft violation scores under activated regimes. The hyperparameters $T_{low},T_{high},\mathrm{and}{C}_{min}$ are set according to empirical quantiles of the standardized dataset (e.g., $T\in [-0.5,0.5],C>-0.3$, which ensures robust regime partitioning and avoids sensitivity to extreme values.

RS-PCVR provides an interpretable measure of physical plausibility:

• RS-PCVR close to 0 indicates that the model prediction respects domain monotonic trends in most activated regimes.
• RS-PCVR close to 1 indicates severe systematic violation, suggesting unreliable physical behavior and poor generalization under out-of-distribution conditions.

Unlike pure accuracy metrics (RMSE/MAE), RS-PCVR reflects whether a model can produce scientifically credible predictions, which is critical for real-world marine monitoring and decision-making applications. In dissolved oxygen prediction tasks, tree-based models such as Random Forest and XGBoost often achieve extremely low RMSE due to strong nonlinear fitting capacity. However, they may generate physically inconsistent gradients, which undermines their reliability in scientific forecasting and environmental management. RS-PCVR complements classical accuracy metrics by providing an additional evaluation dimension: physics plausibility. Therefore, it enables a more comprehensive comparison between black-box regressors and physics-informed neural models.

3.4. Quantitative Comparison Results

3.4.1. Robustness Analysis of Physical Consistency Metrics and Training Stability

Table 1. Predictive Performance and Physical Consistency (PCVR) Under the Default Training Configuration (LR = 1 $\times {10}^{-3}$).

Models	RMSE	MAE	${\mathit{R}}^2$	PCVR
SVR	1.3678	0.6885	0.6827	–
RF	0.1654	0.1026	0.9954	–
XGBoost	0.1811	0.0987	0.9944	–
MLP	0.5348	0.3416	0.9515	0.5187
FINN	0.3130	0.2100	0.9834	0.4826
PI-FINN	0.5132	0.3201	0.9553	0.5092

presents the predictive performance and physical consistency measured by PCVR under the default training configuration with a learning rate of 1 $\times {10}^{-3}$. The results show that RF and XGBoost achieve the best accuracy on this non-temporal structured regression task, where RF obtains an RMSE of 0.165 and an $R^2$ of 0.995, indicating that ensemble learners are highly effective in capturing nonlinear dependencies among marine water-quality variables. Among deep learning baselines, FINN outperforms MLP, reducing RMSE from 0.535 to 0.313 while decreasing PCVR from 0.519 to 0.483, demonstrating the effectiveness of the factor-interaction mechanism in enhancing feature coupling representation and improving physical consistency.

Table 2. Robustness Evaluation of Physical Consistency Metric: Replacing PCVR With RS-PCVR Under a Stable Training Setting (LR = 1 $\times {10}^{-4}$).

Models	RMSE	MAE	${\mathit{R}}^2$	PCVR
SVR	1.3678	0.6885	0.6827	–
RF	0.1654	0.1026	0.9954	–
XGBoost	0.1811	0.0987	0.9944	–
MLP	0.1651	0.1052	0.9733	0.3924
FINN	0.1467	0.0945	0.9789	0.3932
PI-FINN	0.4776	0.3019	0.7764	0.4882

further reports an ablation study building upon Table 1, where the physical consistency metric is replaced from PCVR to RS-PCVR, and the learning rate is reduced from 1 $\times {10}^{-3}$ to 1 $\times {10}^{-4}$ to improve training stability. The results indicate that a smaller learning rate substantially enhances the convergence behavior of deep models: the RMSE of MLP drops from 0.535 to 0.165, and FINN further improves from 0.313 to 0.147. This observation suggests that the default learning rate may lead to unstable optimization or insufficient convergence, while reducing the learning rate yields a smoother training trajectory. Moreover, RS-PCVR provides clearer discrimination of physical consistency across models compared to PCVR. For instance, FINN achieves a lower RS-PCVR of 0.393, whereas PI-FINN remains at 0.488, indicating that RS-PCVR is more robust and less sensitive to noise than the hard-threshold PCVR metric.

Notably, PI-FINN does not outperform FINN even under the RS-PCVR setting, implying that the assumed monotonic physical relationships may not globally hold in real-world marine observations due to environmental heterogeneity and measurement uncertainty. This observation further confirms that PI-FINN does not consistently achieve superior predictive accuracy compared to purely data-driven models. Consequently, directly imposing global physical constraints may cause negative transfer, highlighting the necessity of region-aware and soft physical regularization to avoid over-constraining the learning process. Overall, the incorporation of physical constraints enhances physical consistency and training stability, although it does not necessarily guarantee improvements in predictive accuracy across all scenarios.

3.4.2. Robustness Evaluation of Physical Consistency Metrics Under Different $\lambda$ Scales

Table 3. Prediction Performance and Physical Consistency (PCVR) Under Different Weight Scales $\lambda$ (LR = 1 $\times {10}^{-3}$).

Models	$\mathit{\lambda }$	RMSE	MAE	${\mathit{R}}^2$	PCVR
MLP	0.01	0.1765	0.1089	0.9694	0.5175
	0.05	0.1642	0.1084	0.9736	0.5103
	0.10	0.1581	0.0998	0.9755	0.6012
	0.50	0.1364	0.0906	0.9817	0.4706
	1.00	0.1906	0.1205	0.9644	0.6545
	2.00	0.1919	0.1204	0.9639	0.5291
FINN	0.01	0.1453	0.0936	0.9793	0.5137
	0.05	0.1583	0.0959	0.9754	0.5037
	0.10	0.1577	0.0980	0.9756	0.5497
	0.50	0.1682	0.1048	0.9722	0.6015
	1.00	0.1383	0.0903	0.9812	0.5513
	2.00	0.1637	0.1028	0.9737	0.5734
PI-FINN	0.01	0.2110	0.1325	0.9563	0.5328
	0.05	0.2867	0.1834	0.9194	0.6534
	0.10	0.3474	0.2144	0.8816	0.7085
	0.50	0.4849	0.3106	0.7694	0.7706
	1.00	0.5589	0.3555	0.6937	0.7965
	2.00	0.5103	0.3227	0.7446	0.6578

and

Table 4. Robustness Evaluation Under Different $\lambda$ Scales: Replacing PCVR With RS-PCVR and Using a Stable Training Configuration (LR = 1 $\times {10}^{-4}$).

Models	$\mathit{\lambda }$	RMSE	MAE	${\mathit{R}}^2$	RS-PCVR
MLP	0.01	0.1980	0.1240	0.9616	0.3836
	0.05	0.1935	0.1231	0.9633	0.4032
	0.10	0.1687	0.1068	0.9721	0.4371
	0.50	0.1706	0.1070	0.9715	0.4545
	1.00	0.1651	0.1048	0.9733	0.3437
	2.00	0.1973	0.1285	0.9618	0.4387
FINN	0.01	0.1448	0.0922	0.9794	0.4070
	0.05	0.1449	0.0915	0.9794	0.3550
	0.10	0.1443	0.0923	0.9796	0.4075
	0.50	0.1523	0.0992	0.9773	0.3913
	1.00	0.1478	0.0965	0.9786	0.3832
	2.00	0.1358	0.0886	0.9819	0.4264
PI-FINN	0.01	0.2106	0.1344	0.9565	0.3932
	0.05	0.2989	0.1883	0.9124	0.4841
	0.10	0.3389	0.2163	0.8874	0.5591
	0.50	0.4587	0.2891	0.7937	0.5194
	1.00	0.5138	0.3287	0.7412	0.5290
	2.00	0.5325	0.3286	0.7220	0.4902

report the prediction performance (RMSE, MAE, and $R^2$) and physical consistency evaluation (PCVR/RS-PCVR) of different models under varying weight scales $\lambda$. Notably, $\lambda$ is defined identically in both experiments, representing the weighting coefficients of three driving variables, i.e., temperature (T), depth (D), and chlorophyll concentration (C), while different magnitudes of $\lambda$ are tested to investigate the scale sensitivity of the proposed physical consistency metrics.

The results indicate that both PCVR and RS-PCVR vary significantly with $\lambda$, demonstrating that physical consistency evaluation is highly dependent on the assumed relative importance of different driving factors. This observation is consistent with real marine environments, where the dominant factors affecting dissolved oxygen dynamics may change across different regions and conditions. Compared with other models, FINN shows smaller fluctuations of PCVR/RS-PCVR across a wide range of $\lambda$ values, suggesting stronger robustness and more balanced learning of multi-factor driven DO trends. In contrast, PI-FINN exhibits more noticeable degradation under certain $\lambda$ settings, implying higher sensitivity to the evaluation weighting scale.

Furthermore, comparing Table 3 (learning rate 1 $\times {10}^{-3}$) and Table 4 (learning rate 1 $\times {10}^{-4}$) reveals that a smaller learning rate improves the prediction accuracy of deep models, especially for FINN, indicating the importance of optimization stability for marine DO regression tasks. Meanwhile, RS-PCVR demonstrates smoother variations than PCVR under different $\lambda$ magnitudes, verifying that RS-PCVR provides a more robust and reliable physical consistency assessment in noisy marine observations. It should be noted that $\lambda$ is only used in the evaluation stage and does not participate in model training.

3.4.3. Sensitivity Analysis of PCVR Under Different Driving-Factor Combinations

To investigate the sensitivity of physical consistency evaluation under different driving factors, we decompose PCVR/RS-PCVR into three components corresponding to temperature (T), depth (D), and chlorophyll concentration (C), and evaluate multiple combinations (T-only, D-only, C-only, pairwise combinations, and T/D/C jointly).

Table 5. Model Performance and PCVR Under Different Driving-Factor Activation Combinations (LR = 1 $\times {10}^{-3}$). Note: ✓ indicates that the corresponding driving-factor consistency term is activated.

Models	${\mathit{\lambda }}_{\mathit{T}}$	${\mathit{\lambda }}_{\mathit{D}}$	${\mathit{\lambda }}_{\mathit{C}}$	RMSE	MAE	${\mathit{R}}^2$	PCVR
MLP				0.5348	0.3416	0.9515	0.5187
	✓			0.1700	0.1047	0.9717	0.4432
		✓		0.2020	0.1270	0.9600	0.4687
			✓	0.1536	0.0974	0.9769	0.5025
	✓	✓		0.1716	0.1063	0.9711	0.4277
	✓		✓	0.1793	0.1149	0.9685	0.5054
		✓	✓	0.2076	0.1325	0.9577	0.4909
	✓	✓	✓	0.1906	0.1205	0.9644	0.6545
FINN				0.3130	0.2100	0.9834	0.4826
	✓			0.1461	0.0943	0.9791	0.5181
		✓		0.1515	0.0947	0.9775	0.5254
			✓	0.1571	0.0964	0.9758	0.6502
	✓	✓		0.1262	0.0828	0.9844	0.5787
	✓		✓	0.1527	0.0971	0.9771	0.5520
		✓	✓	0.1287	0.0850	0.9838	0.5641
	✓	✓	✓	0.1383	0.0903	0.9812	0.5513
PI-FINN				0.5132	0.3201	0.9553	0.5092
	✓			0.3649	0.2112	0.8694	0.6207
		✓		0.1283	0.0834	0.9839	0.5271
			✓	0.1840	0.1205	0.9668	0.5096
	✓	✓		0.4085	0.2526	0.8364	0.5890
	✓		✓	0.4806	0.3061	0.7735	0.6802
		✓	✓	0.1890	0.1242	0.9650	0.5766
	✓	✓	✓	0.5589	0.3555	0.6937	0.7965

and

Table 6. Sensitivity of RS-PCVR Under Different T/D/C Factor Combinations With Improved Training Stability (LR = 1 $\times {10}^{-4}$). Note: ✓ indicates that the corresponding driving-factor consistency term is activated.

Models	${\mathit{\lambda }}_{\mathit{T}}$	${\mathit{\lambda }}_{\mathit{D}}$	${\mathit{\lambda }}_{\mathit{C}}$	RMSE	MAE	${\mathit{R}}^2$	RS-PCVR
MLP				0.1651	0.1052	0.9733	0.3924
	✓			0.1638	0.1082	0.9737	0.4124
		✓		0.1686	0.1071	0.9721	0.4256
			✓	0.2057	0.1318	0.9585	0.4011
	✓	✓		0.1565	0.0975	0.9760	0.3794
	✓		✓	0.1716	0.1080	0.9711	0.3637
		✓	✓	0.1836	0.1187	0.9669	0.4085
	✓	✓	✓	0.1651	0.1048	0.9733	0.3437
FINN				0.1467	0.0945	0.9789	0.3932
	✓			0.1447	0.0929	0.9795	0.4178
		✓		0.1481	0.0926	0.9785	0.3667
			✓	0.1448	0.0907	0.9794	0.3926
	✓	✓		0.1549	0.0952	0.9765	0.4237
	✓		✓	0.1432	0.0908	0.9799	0.3963
		✓	✓	0.1370	0.0886	0.9816	0.3666
	✓	✓	✓	0.1478	0.0965	0.9786	0.3832
PI-FINN				0.4776	0.3019	0.7764	0.4882
	✓			0.3873	0.2304	0.8529	0.4496
		✓		0.1545	0.0961	0.9766	0.3862
			✓	0.1996	0.1277	0.9609	0.4354
	✓	✓		0.3931	0.2401	0.8485	0.4658
	✓		✓	0.4930	0.3193	0.7617	0.5446
		✓	✓	0.2329	0.1462	0.9468	0.3890
	✓	✓	✓	0.5138	0.3287	0.7412	0.5290

present the results under learning rates of 1 $\times {10}^{-3}$ and 1 $\times {10}^{-4}$, respectively, while all other configurations remain unchanged. PCVR and RS-PCVR is only used as an evaluation metric and is not involved in training; thus, different T/D/C weight combinations do not change model predictions but only affect the physical consistency score.

The results demonstrate that PCVR/RS-PCVR is highly sensitive to the selected driving-factor combination. When only a single driving factor is considered, the violation rate is generally lower; however, when T, D, and C are jointly activated, PCVR/RS-PCVR increases substantially across all models. This indicates that real-world marine DO observations do not always satisfy monotonic consistency assumptions with respect to all driving factors simultaneously, and multi-factor coupling conditions may lead to cumulative violations. This phenomenon highlights the environment-dependent nature of physical consistency assessment and suggests that a single fixed physical rule may not be sufficient to describe complex marine oxygen dynamics.

From the model comparison perspective, FINN maintains relatively stable and low prediction errors across different driving-factor combinations, suggesting stronger robustness and balanced learning capability under multi-variable interactions. In contrast, PI-FINN exhibits larger fluctuations in both error metrics and PCVR/RS-PCVR, particularly under the T/D/C joint setting, implying that naive physical prior injection may introduce negative transfer and lead to biased fitting toward specific driving assumptions in noisy marine environments.

Moreover, comparing Table 5 (learning rate 1 $\times {10}^{-3}$) and Table 6 (learning rate 1 $\times {10}^{-4}$) reveals that a smaller learning rate significantly improves the prediction accuracy of deep models (especially MLP and FINN) and yields lower PCVR/RS-PCVR under most factor combinations, indicating that optimization stability plays a crucial role in DO regression tasks and that improved prediction accuracy can indirectly enhance trend consistency.

3.4.4. Training-Stage Robustness of RS-PCVR for Physical Consistency Monitoring

As illustrated in Figure 1a–f, the training dynamics of physical-consistency metrics exhibit clear model-dependent convergence patterns under different weighting configurations $\lambda$. Specifically, the PCVR curves for MLP, FINN, and PI-FINN generally show a “decrease–fluctuation–stabilization” behavior during the training process. In most cases, PCVR declines rapidly in the early epochs and then gradually converges to a stable plateau, although the convergence level and oscillation magnitude differ significantly among models. In contrast, the RS-PCVR curves display a more consistent “increase–stabilization” trend, where RS-PCVR steadily improves throughout training and reaches a stable region with noticeably reduced variance.

Furthermore, when comparing different $\lambda$ combinations (corresponding to different weight allocations for T, D, and C), both PCVR and RS-PCVR demonstrate distinct sensitivity to the weighting strategy. Under small $\lambda$ settings, PCVR often exhibits more pronounced degradation and oscillations, whereas larger $\lambda$ values lead to smoother convergence trajectories and improved stability. Across all $\lambda$ configurations, PI-FINN consistently maintains more stable metric trajectories than MLP and FINN, suggesting enhanced robustness in terms of physical consistency.

The observed discrepancy between PCVR and RS-PCVR trends can be attributed to their inherent evaluation characteristics. PCVR is directly affected by the absolute magnitude of physical residual violations, which makes it more sensitive to prediction amplitude changes, outliers, and local instability during parameter updates. Consequently, during the early stage of training, when the model rapidly adapts to the data distribution, PCVR tends to decrease sharply and may exhibit oscillations as the prediction distribution shifts.

On the other hand, RS-PCVR adopts a relatively normalized and robust measurement formulation, which mitigates the impact of scale mismatch and extreme residual values. Therefore, RS-PCVR can better capture the gradual improvement of physical consistency during model convergence and presents a smoother monotonic trend.

In addition, since PCVR and RS-PCVR are not explicitly included in the loss function (i.e., they are not optimized through backpropagation), their convergence behaviors reflect an implicit outcome of prediction refinement rather than a direct physical-constraint enforcement process. Under this setting, the learning rate becomes a dominant factor: a smaller learning rate reduces parameter update amplitude, which naturally suppresses metric oscillations and improves convergence stability.

Based on the above results, three key conclusions can be drawn: (1) PCVR and RS-PCVR exhibit fundamentally different convergence characteristics, where PCVR tends to decrease and stabilize, while RS-PCVR generally increases and stabilizes, indicating that RS-PCVR provides a more stable evaluation perspective for training-stage physical consistency; (2) RS-PCVR demonstrates stronger robustness and smoother convergence than PCVR, with reduced sensitivity to oscillations induced by model parameter updates, especially in the mid-to-late training stage; (3) PI-FINN consistently achieves the most stable convergence behavior across different $\lambda$ settings, characterized by smaller fluctuations and more consistent plateau values, outperforming both FINN and the purely data-driven MLP baseline. This suggests that the PI-FINN architecture is inherently more capable of maintaining stable physically consistent predictions during iterative optimization.

These findings highlight that physical consistency should not only be evaluated by final converged values but also by its training-stage convergence stability, since unstable physical-consistency evolution may indicate sensitivity to hyperparameter perturbations and reduced generalization reliability.

Moreover, the proposed RS-PCVR metric provides a more robust and interpretable tool for monitoring physical-consistency evolution throughout training, making it suitable for comparative analysis under different training configurations. Finally, the superior convergence stability observed in PI-FINN implies that physics-inspired structural design can effectively improve robustness against training instability, offering practical advantages for real-world dissolved oxygen prediction tasks where physical plausibility and generalization performance are both critical.

4. Discussion

This study investigates dissolved oxygen (DO) prediction in marine environments under a non-temporal learning paradigm, where DO is estimated solely from four driving variables, namely depth, temperature, salinity, and chlorophyll concentration. Comprehensive experiments are conducted to compare conventional machine learning baselines (SVR, RF, XGBoost), data-driven neural models (MLP), and physics-inspired architectures (FINN and PI-FINN). In addition, PCVR and its enhanced variant RS-PCVR are introduced as physics-consistency evaluation metrics, and their convergence behaviors under different weight configurations $\lambda$ are systematically analyzed, providing a new perspective for assessing model stability and physical reliability. The proposed framework further demonstrates practical potential for environmental decision-making by delivering physically consistent DO estimates under sparse observation conditions. Such capability is particularly valuable for identifying hypoxic regions, supporting adaptive monitoring strategies, and informing early warning systems.

From a sustainability perspective, the proposed framework provides a practical tool for improving the reliability of environmental monitoring systems under data-limited conditions. Physically consistent predictions of dissolved oxygen are essential for detecting hypoxia risks, supporting ecosystem protection, and enabling informed decision-making in water resource management. This contributes to sustainability by enhancing the scientific basis for environmental assessment and long-term ecological management.

4.1. Performance Gap Between Classical ML and Neural Physics-Inspired Models

The experimental results show that ensemble-based machine learning methods, particularly RF and XGBoost, achieve remarkably strong predictive accuracy, with RMSE values in the range of 0.16∼0.18 and $R^2$ exceeding 0.99. In contrast, the MLP baseline exhibits a substantially higher RMSE, while FINN improves upon MLP but still does not surpass RF/XGBoost in terms of pure numerical accuracy. Notably, PI-FINN does not consistently outperform FINN and, in several cases, even shows degraded RMSE and MAE performance.

This phenomenon can be attributed to both the characteristics of the dataset and the underlying modeling assumptions. Since the task explicitly excludes temporal dependencies, the problem reduces to a static regression mapping between environmental variables and DO. In such a setting, tree-based ensemble methods (RF/XGBoost) are well known to be highly competitive, as they can effectively capture complex nonlinear interactions, accommodate heteroscedasticity, and remain robust to outliers as well as feature-scale inconsistencies. Of course, the extremely high R² values observed in tree-based models may indicate potential overfitting. In contrast, neural models typically require more careful hyperparameter tuning, feature normalization, and training stabilization, particularly when the architecture involves multiple interaction branches or physics-guided components.

Therefore, the superior performance of RF/XGBoost suggests that the dataset contains highly learnable nonlinear structures that can be effectively approximated through purely data-driven statistical fitting. This observation further indicates that, under the current experimental setting, the primary advantage of physics-inspired neural networks may not lie in marginal gains in RMSE, but rather in improved interpretability, training stability, and physical plausibility. In particular, restricting the model to second-order interactions enhances interpretability and structural transparency, but may simultaneously limit its capacity to capture higher-order nonlinear dependencies present in complex marine environments. The performance gap between MLP and FINN further indicates that the explicit interaction decomposition contributes substantially to the observed improvements. This suggests that FINN benefits not only from neural nonlinear approximation capacity but also from its structured interaction-aware design.

4.2. Why PI-FINN Does Not Always Outperform FINN

A critical finding of this study is that PI-FINN does not consistently guarantee superior performance compared to FINN, and its effectiveness is sensitive to training configurations. Several factors can account for this behavior.

First, PI-FINN introduces additional structural constraints that effectively reduce the hypothesis space. While such constraints may improve generalization when the embedded physical assumptions are well aligned with the underlying system, they can also introduce inductive bias if the assumed mechanisms are incomplete or partially inconsistent with the data distribution. In real marine environments, DO dynamics are governed by complex and highly coupled processes (e.g., mixing, photosynthesis, respiration, stratification, and biological consumption), many of which are not explicitly captured in a simplified physics-informed formulation. Consequently, the physics-guided branch may restrict model flexibility and limit its capacity to approximate highly nonlinear relationships, potentially leading to degraded predictive performance.

Second, PI-FINN involves more complex gradient propagation pathways due to the incorporation of physics-based regularization terms. Without appropriate feature standardization and carefully designed training strategies (e.g., staged optimization or weight scheduling), the learning process may become unstable. This is consistent with the empirical observation that PCVR and RS-PCVR curves often exhibit oscillatory or plateauing behavior, particularly under relatively large weight settings. Such instability indicates that PI-FINN is more sensitive to hyperparameters such as learning rate and parameter initialization, increasing the likelihood of convergence to suboptimal local minima.

Therefore, PI-FINN should primarily be viewed as a physically guided reliability-enhancement framework rather than a purely accuracy-oriented predictive model whose effectiveness depends on both the validity of the assumed physical mechanisms and the stability of the optimization process. In this context, the primary role of PI-FINN is to enforce physical consistency in model predictions rather than to directly maximize predictive accuracy. Accordingly, PI-FINN prioritizes physical consistency and robustness over purely predictive performance, especially in scenarios where physical plausibility is of primary concern.

4.3. Effects of Learning Rate on Metric Convergence Stability

Another important observation is that reducing the learning rate from $1\times {10}^{-3}$ to $1\times {10}^{-4}$ significantly improves the stability of PCVR/RS-PCVR curves across models and weight configurations. Under the smaller learning rate, the reported RMSE and $R^2$ values become more consistent across different $\lambda$ settings, and the fluctuations of physical-consistency metrics are reduced.

This result indicates that PCVR and RS-PCVR are not only indicators of physical plausibility but also sensitive diagnostic tools reflecting training dynamics. Since PCVR/RS-PCVR are used purely as evaluation metrics rather than loss constraints, their convergence behavior is implicitly determined by the smoothness of prediction updates. A high learning rate causes larger parameter jumps and stronger oscillations in prediction distributions, which amplifies local physical inconsistency and increases metric variance. In contrast, a smaller learning rate leads to smoother parameter evolution and more stable physical-consistency trajectories, which is particularly beneficial for PI-FINN due to its complex multi-branch structure.

This finding also implies that training stability is a necessary prerequisite before evaluating whether physics-inspired architectures truly improve physical reliability.

4.4. Interpretation of Opposite Trends Between PCVR and RS-PCVR

The convergence analysis reveals a consistent pattern: PCVR tends to decrease rapidly in early epochs and then stabilizes, whereas RS-PCVR exhibits a gradual increasing trend with smoother convergence. This opposite behavior reflects the fundamental difference between the two metrics.

PCVR directly measures absolute physical violation frequency or magnitude. In the early training stage, the model quickly reduces large prediction errors and moves outputs into more feasible physical ranges, leading to an immediate decrease in violations. However, later-stage fine-tuning may focus on fitting small-scale nonlinearities and noise patterns, which can reintroduce local inconsistency and cause PCVR to plateau or oscillate.

RS-PCVR, in contrast, evaluates physical consistency from a normalized or relative perspective. Instead of focusing solely on whether the output violates an absolute constraint, it measures whether the model prediction preserves physically meaningful response patterns under varying temperature, depth, and chlorophyll regimes. Since DO is governed by stable response structures (e.g., temperature-dependent solubility decrease and stratification-induced oxygen depletion), the gradual improvement of RS-PCVR suggests that the model increasingly learns physically interpretable relationships beyond pure numerical fitting.

Therefore, RS-PCVR provides complementary information: PCVR emphasizes feasibility, while RS-PCVR emphasizes structural physical interpretability. This explains why RS-PCVR can be considered more suitable for convergence stability analysis and for evaluating whether a model truly aligns with physical mechanisms.

4.5. Sensitivity to $\lambda$ Configurations and the Role of Driving Variables

The experiments further demonstrate that the weight configuration $\lambda$, which controls the relative contributions of temperature T, depth D, and chlorophyll C in the PCVR/RS-PCVR computation, has a measurable impact on both the resulting metric values and their convergence behavior. In general, assigning nonzero weights to multiple driving variables simultaneously tends to increase PCVR values, indicating that enforcing multi-factor physical consistency is more challenging than satisfying single-factor constraints. This observation is consistent with the fact that DO dynamics are governed by multiple interacting processes, and achieving consistency across all drivers requires the model to learn a more globally coherent mapping. In particular, dissolved oxygen profiles may exhibit abrupt vertical variations due to stratification effects, and the relationship between chlorophyll concentration and dissolved oxygen is inherently complex and not strictly monotonic across different environmental regimes.

Additionally, the results suggest that different models exhibit distinct sensitivities under identical $\lambda$ configurations. FINN generally produces more stable PCVR/RS-PCVR trends compared to MLP, implying that its structured interaction design introduces an implicit regularization effect. In contrast, PI-FINN, while offering enhanced physical interpretability, demonstrates increased sensitivity under certain multi-driver combinations. This behavior supports the interpretation that physics-informed components introduce additional optimization complexity and require more careful tuning.

From an oceanographic perspective, this sensitivity analysis provides meaningful insights: it suggests that DO prediction models may respond differently under varying dominant environmental regimes, such as temperature-driven surface variability or depth-induced stratification. Therefore, analyzing PCVR/RS-PCVR across different $\lambda$ configurations offers an interpretable framework for assessing model robustness and physical consistency under heterogeneous marine conditions.

4.6. Implications for Real-World DO Monitoring and Algorithmic Contribution

Although RF and XGBoost outperform deep models in pure RMSE and $R^2$, the proposed evaluation framework demonstrates that accuracy alone is insufficient for reliable marine DO monitoring. In real applications such as hypoxia risk warning, marine ecosystem protection, and intelligent ocean observation systems, prediction reliability depends not only on numerical fit but also on whether the predicted DO values behave consistently with physical mechanisms.

The proposed RS-PCVR metric and the convergence stability analysis provide a practical solution to this issue. By evaluating physical consistency dynamically across training and across different driving-variable weight configurations, the framework enables a more comprehensive assessment of model reliability and interpretability. This is particularly valuable when deploying data-driven models in environmental monitoring systems, where unseen distribution shifts and noisy measurements are common.

Therefore, the key contribution of this work is not limited to proposing a physics-inspired model architecture, but rather establishing a new evaluation paradigm for marine DO prediction: combining conventional accuracy metrics with RS-PCVR-based physical consistency assessment and convergence stability analysis. This paradigm provides a robust and extensible methodology that can be generalized to other oceanographic variables and environmental prediction tasks.

4.7. Limitations and Future Work

The proposed framework is subject to several limitations, including the absence of uncertainty quantification, simplified physical assumptions, sensitivity to hyperparameter configurations, and limited validation across diverse settings. First, the current statistical formulation remains limited, and discrepancies may exist between the empirical modeling strategy and the underlying physical processes. In particular, the model implicitly assumes homoscedastic errors and does not explicitly account for non-Gaussian data distributions. Furthermore, another limitation of the current study is the absence of explicit uncertainty quantification. Although the proposed framework evaluates physical consistency and convergence stability, predictive uncertainty remains unexplored. Future work may incorporate bootstrap confidence intervals, Bayesian neural estimation, or ensemble-based uncertainty analysis to improve the reliability assessment of dissolved oxygen prediction under sparse observational conditions.

Second, the model does not explicitly incorporate temporal dynamics, which may restrict its applicability in environments characterized by strong temporal variability. By design, the proposed framework omits temporal dependencies, thereby limiting its ability to capture short-term environmental fluctuations such as storms, mixing events, and upwelling processes. These episodic phenomena can induce abrupt variations in dissolved oxygen that are not fully explained by instantaneous environmental variables alone. Consequently, the current model is better suited for capturing relatively stable environmental relationships rather than transient dynamics. Future work may explore hybrid modeling strategies that incorporate event-level or sparse temporal information without relying on fully continuous time-series data.

It should also be noted that the present study is conducted using a single regional marine dataset collected from Rongcheng Bay. Although the proposed framework demonstrates promising performance under sparse observational conditions, its generalization capability across different oceanographic regimes, climatic conditions, and monitoring systems remains to be further validated. Future work will therefore investigate cross-region transferability and domain adaptation under heterogeneous marine environments.

Therefore, the generalization capability of FINN and PI-FINN beyond the training region remains an open question. In addition, the model may exhibit sensitivity to hyperparameter configurations, especially under sparse data conditions, which can affect both convergence stability and predictive performance. These limitations should be carefully considered when deploying the model in real-world applications. Future work will further extend the framework to multi-regional datasets and investigate domain adaptation techniques to enhance model transferability across heterogeneous marine environments.

5. Conclusions

This paper proposes a non-temporal dissolved oxygen (DO) prediction framework that estimates DO concentration directly from environmental drivers without explicitly relying on time-series dependencies. To address the nonlinear and coupled relationships among depth, salinity, temperature, and chlorophyll-a concentration, we develop a Factor-Interaction Neural Network (FINN), which explicitly decomposes DO formation into individual factor contributions and structured pairwise interaction effects. Such an interaction-driven architecture enables the model to capture complex cross-factor dependencies while maintaining improved interpretability compared with conventional black-box neural regressors.

Building upon FINN, we further propose a physics-informed extension, namely PI-FINN, by incorporating oceanographic-consistent priors that reflect key physical mechanisms in DO variation, including temperature-related solubility behavior, depth-dependent stratification effects, and chlorophyll-associated biological oxygen production patterns. In addition to standard accuracy metrics, we introduce a physics-consistency evaluation protocol based on physically meaningful violation-rate measures and conduct a systematic convergence and stability analysis under different driver-weight configurations. This provides a practical diagnostic tool for assessing whether a predictive model exhibits physically plausible responses across heterogeneous marine regimes.

Extensive experiments and ablation studies demonstrate that explicitly modeling factor interactions improves DO prediction performance and training robustness. Moreover, the physics-informed design of PI-FINN contributes to enhanced physical plausibility and stability, particularly under noisy observations and varying environmental conditions. These results indicate that the proposed framework offers a promising approach for integrating data-driven modeling and physical consistency, contributing to sustainable environmental monitoring and management under data-limited conditions. Overall, the results highlight a fundamental trade-off between predictive accuracy and physical consistency: while purely data-driven models may achieve higher numerical accuracy, they are more prone to violating physical constraints, whereas PI-FINN explicitly prioritizes physical consistency and robustness over purely predictive performance. Consequently, the practical applicability of the proposed framework depends on the validity of the underlying physical assumptions and the target application requirements. Overall, PI-FINN mainly contributes to enhancing physical plausibility, robustness, and interpretability under sparse monitoring conditions, rather than consistently improving conventional prediction-error metrics. Of course, the general applicability of FINN and PI-FINN across different marine environments still requires further validation using multi-region datasets.

Future work will place greater emphasis on environmental interpretability and domain-specific analysis, aiming to further bridge the gap between data-driven modeling and physical understanding. Meanwhile, we focus on extending the proposed framework toward broader generalization by incorporating spatiotemporal context, multi-depth vertical profile structures, and uncertainty-aware prediction, thereby enabling risk-sensitive DO forecasting and real-time decision support in practical ocean monitoring systems.