DeepChance-OPT: A Robust Decision-Making Framework for Dynamic Grasping in Precision Assembly

Abstract

Achieving safe and efficient sequential decision-making in dynamic and uncertain environments is a core challenge in intelligent manufacturing and robotic systems. During operation, systems are often subject to coupled multi-source uncertainties—such as stochastic disturbances, model mismatch, and environmental shifts—rendering traditional approaches based on deterministic models or post hoc safety verification incapable of simultaneously ensuring performance and safety. In particular, the non-differentiability of constraint satisfaction probabilities in chance-constrained decision-making severely impedes its integration with data-driven learning paradigms. To address these challenges, this paper proposes DeepChance-OPT (Deep Chance Optimization), an end-to-end differentiable disturbance-rejection decision framework tailored for dynamic grasping tasks in precision assembly. The framework first encodes historical observations and control sequences into a low-dimensional latent representation to extract key dynamic features relevant to decision-making. Subsequently, it models the temporal propagation of uncertainty in this latent space to predict the probability distribution of future states. Furthermore, via a differentiable chance-constrained mechanism, the risk of constraint violation is transformed into a continuous and differentiable penalty term, which is jointly optimized with the task performance objective to achieve synergistic improvement in both safety and efficiency. The entire framework is trained and executed under a unified end-to-end architecture, enabling closed-loop online sequential decision-making. Experiments on a precision silicon carbide wafer grasping task demonstrate that DeepChance-OPT achieves real-time performance (average decision latency < 4 ms) while reducing the constraint violation rate to 2.3%, significantly outperforming both traditional optimization and purely data-driven baselines. Under composite uncertainty scenarios—including parameter perturbations, measurement noise, and external disturbances—the success rate remains stably above 87.5%, fully validating the effectiveness of the proposed framework for robust, safe, and efficient decision-making in complex dynamic environments. This work provides a new paradigm for intelligent disturbance-rejection decision-making in high-precision manufacturing, offering both theoretical rigor and engineering practicality.

1. Introduction

In the fields of intelligent manufacturing, robotic manipulation, and autonomous systems, safe sequential decision-making in complex dynamic environments has become a central research challenge. In practical tasks, an agent must generate control policies in real time that are both efficient and satisfy stringent safety constraints under the coupled effects of multi-source uncertainties—such as stochastic disturbances, model mismatch, and environmental shifts [1]. However, conventional approaches face a dual dilemma: on one hand, designs based on deterministic models or static safety margins struggle to adapt to dynamically evolving risks; on the other hand, existing learning-based decision paradigms, despite their strong representational capacity, often treat safety as a post hoc verification step or a heuristic penalty, lacking explicit modeling of constraint violation probabilities [2,3]. This leads to severe degradation in safety performance under distributional shifts or rare perturbations.

Chance-constrained optimization (CCO) [4,5] offers a probabilistic theoretical guarantee for safe decision-making by bounding the probability of constraint violation. However, its practical deployment is hindered by two major bottlenecks: first, it heavily relies on accurate analytical system models, resulting in prohibitive modeling costs in high-dimensional nonlinear settings; second, the indicator function underlying chance constraints is typically non-differentiable and discontinuous, making it difficult to integrate with deep learning frameworks in an end-to-end manner [6,7]. Consequently, CCO remains largely decoupled from modern data-driven policy learning pipelines, limiting its applicability in intelligent systems that require adaptive and real-time responses.

Critically, uncertainty in dynamic environments does not exist in isolation [2]; rather, it propagates over time and couples intricately with state evolution and control inputs. Ignoring this temporal evolution renders risk assessment lagging behind the decision process, potentially triggering safety hazards. Recent advances in uncertainty quantification and propagation—such as surrogate modeling with Gaussian processes [8], Bayesian neural networks [9], and physics-informed deep learning [10]—highlight the feasibility of online uncertainty forecasting. However, these methods rarely interface directly with constrained decision layers in a differentiable fashion. Therefore, there is an urgent need for a decision-making mechanism capable of online perception, propagation, and response to uncertainty information, unifying proactive risk modeling and performance optimization within a single information processing pipeline.

To this end, we propose DeepChance-OPT—an end-to-end differentiable disturbance-rejection decision framework tailored for dynamic grasping tasks in precision assembly. The framework achieves safe and efficient sequential decision-making through a three-stage information processing pipeline: (1) leveraging latent representation learning to extract low-dimensional dynamic features from historical observations and control sequences; (2) constructing an adaptive uncertainty propagation model in the latent space to predict the probability distribution of future states, thereby providing statistical priors for risk assessment; and (3) introducing a differentiable chance-constrained mechanism that translates safety requirements into a continuous and differentiable optimization objective, enabling synergistic optimization of performance and safety. The entire framework is trained and executed under a unified end-to-end architecture, supporting closed-loop online decision-making.

We systematically evaluate DeepChance-OPT on a representative industrial task: precision grasping of silicon carbide wafers. Experimental results show that the method achieves real-time performance (average latency < 4 ms) while reducing the constraint violation rate to 2.3%, significantly outperforming both traditional optimization and purely learning-based baselines. Under composite perturbations, it maintains a grasping success rate above 87.5%, fully validating its robustness and practicality.

The main contributions of this work are summarized as follows:

• We propose DeepChance-OPT, the first disturbance-rejection decision framework that unifies temporal uncertainty propagation modeling, differentiable chance constraints, and end-to-end sequential decision-making.
• We design a latent-representation-based adaptive uncertainty propagation mechanism that enables proactive quantification of future risk.
• We develop a differentiable chance-constrained modeling approach, achieving, for the first time, end-to-end joint optimization between probabilistic safety constraints and deep policy networks.
• We validate the proposed framework on a real-world precision assembly task, demonstrating its comprehensive advantages in safety, performance, and real-time capability, thereby offering a new paradigm for intelligent decision-making in high-precision manufacturing.

2. Related Work

This work intersects four interrelated research directions in safe sequential decision-making under uncertainty.

Uncertainty Optimization and Chance-Constrained Decision-Making. Chance-constrained programming (CCP), pioneered by Charnes and Cooper [1], provides a foundational framework for ensuring probabilistic feasibility under stochasticity. Miller and Wagner [11] extended CCP to joint chance constraints. With advances in computation, data-driven approaches have gained prominence: Geng and Xie [12,13] comprehensively reviewed scenario-based methods, sample average approximation (SAA), and robust optimization techniques. Küçükyavuz and Jiang [5] surveyed reformulations under limited distributional information. Recently, distributionally robust chance-constrained programming (DRCCP) has attracted significant attention due to its resilience against distributional ambiguity. Representative formulations leverage Wasserstein distance [14,15], relative entropy [16], and polyhedral ambiguity sets [17]. DRCCP has been successfully applied to power systems [4,16], facility location [14], and humanitarian logistics [18].

Uncertainty Modeling and Propagation. Uncertainty quantification (UQ) and propagation are prerequisites for risk-aware decision-making. While traditional Monte Carlo methods are computationally prohibitive, modern UQ emphasizes efficient propagation. Zhang [19] surveyed accelerated techniques like multilevel Monte Carlo. In the deep learning era, Bayesian neural networks [9], Gaussian processes [8,20], and normalizing flows [21] are employed to construct probabilistic surrogates that explicitly output predictive distributions. Notably, frameworks such as RBA [9] and LVM-GP [8] enable end-to-end uncertainty propagation, while Diamzon and Venturi [22] derived analytical propagation formulas for leaky ReLU networks. Physics-informed deep learning has also been used for uncertainty propagation in geophysical inversion [10]. Collectively, these methods drive a paradigm shift from point prediction to distributional prediction.

Data-Driven Safe Decision-Making. In safety-critical domains like autonomous driving and robotics, translating uncertainty into safe behaviors remains a core challenge. Zhou et al. [23] proposed a knowledge-and-data-driven, risk-sensitive framework using large language models for high-risk scenarios. Mazumdar et al. [24] introduced data-driven robust safety verification for Markov decision processes with high-confidence guarantees. Reinforcement learning approaches have also been explored: Zhu et al. [25] designed a DDPG-based policy verified via formal methods; Yuan et al. [26] constructed a safety envelope to constrain DRL exploration. Additionally, conformal prediction [27] and chance constraint learning [7] inject probabilistic safety guarantees into optimization. System-theoretic safety analysis methods like STPA have been extended to learning-enabled systems [28], providing theoretical grounding for safe decision-making.

Integrated Learning-Optimization Frameworks. To overcome the suboptimality of two-stage “predict-then-optimize” pipelines, researchers advocate end-to-end differentiable architectures. Wang et al. [29] reviewed sequential optimization, end-to-end learning, and direct learning paradigms. Differentiable optimization layers (e.g., BPQP [6]) enable gradient backpropagation through optimization subproblems for joint training. Decision-focused learning (DFL) [2] directly optimizes prediction models using decision loss. Yang et al. [3] proposed decision-focused fine-tuning (DFF) to balance prediction accuracy and decision performance under limited data. In applications, Zhang et al. [30] developed a cloud-edge-device collaborative closed-loop framework for smart manufacturing, while Shahnawaz and Safder [31] integrated Bayesian learning with stochastic optimization for adaptive supply chain configuration. These efforts converge on a trend: deeply integrating learning, inference, and optimization into a unified “perception-decision-execution-feedback” intelligent loop. However, existing studies typically address uncertainty modeling, chance-constrained optimization, or learning-based decision-making as separate components, and often rely on either static uncertainty assumptions or post-hoc safety verification. In contrast, DeepChance-OPT explicitly fills this gap by providing an end-to-end differentiable framework that tightly integrates temporal uncertainty propagation, differentiable chance constraints, and online sequential decision-making, enabling joint optimization of performance and safety under dynamically evolving uncertainty.

3. Methodology

3.1. Problem Formulation and Learning Objective

This work addresses sequential decision-making under multimodal and time-varying uncertainties. We consider a dynamical system evolving in discrete time steps t, whose dynamics are unknown and potentially nonlinear, while the decision-maker only has access to partial and noisy observations.

Let $x_t\in {\mathbb{R}}^n$ denote the true system state at time t, and $u_t\in {\mathbb{R}}^m$ the corresponding control input. Due to sensor noise, environmental disturbances, and model mismatch, the true state $x_t$ is not directly observable; instead, the agent receives an observation $y_t\in {\mathbb{R}}^p$, which constitutes an incomplete and corrupted mapping of the underlying state. We define a history window of length T as

$${\mathcal{H}}_t=\{y_{t-T+1},\dots ,y_t,u_{t-T+1},\dots ,u_{t-1}\}$$

(1)

which captures the temporal behavior of the system and serves as the primary input for subsequent prediction and decision-making.

The system evolution is subject to multi-source uncertainties—including stochastic disturbances, model mismatch, and latent environmental shifts—which often exhibit multimodal and non-stationary characteristics. These properties render analytical probabilistic modeling intractable. Consequently, the future state evolution should be represented not as a single trajectory but as a probability distribution.

Under this setting, our objective is to compute a sequence of control inputs over a finite prediction horizon N that minimizes the expected performance cost while satisfying safety constraints. Formally, the problem is stated as:

$$\underset{\{u_t,\dots ,u_{t+N-1}\}}{min}E\left[\sum _{k=0}^{N-1}\mathcal{l}(x_{t+k},u_{t+k})\right]$$

(2)

subject to the following chance constraints:

$$\mathbb{P}\left(g\left(x_{t+k}\right)\le 0\right)\ge 1-\epsilon ,\forall k=1,\dots ,N,$$

(3)

where $\mathcal{l}(·)$ denotes the stage cost function, $g(·)$ represents system safety or physical constraints, and $\epsilon \in (0,1)$ is the maximum allowable constraint violation probability.

Solving this problem entails several challenges: First, the high complexity and time-varying nature of system uncertainties make traditional analytical modeling and inference approaches inapplicable. Second, classical chance-constrained optimization methods suffer from high computational complexity and are difficult to integrate end-to-end with data-driven models. Third, safety constraints must be satisfied in real time during online decision-making, imposing stringent requirements on computational efficiency and robustness.

To address these issues, we aim to construct a data-driven decision-learning framework that explicitly models the propagation of uncertainty and embeds chance constraints in a differentiable manner into the joint learning-and-optimization pipeline. The learning objective is to simultaneously enhance decision performance and safety reliability under both end-to-end training and online inference settings.

3.2. Framework Overview: DeepChance-OPT

To address the challenge of safe sequential decision-making under multimodal and time-varying uncertainties, we propose DeepChance-OPT, an end-to-end differentiable framework that unifies learning and optimization. This framework organically integrates uncertainty perception, propagation modeling, chance-constrained handling, and online decision optimization into a closed-loop information processing pipeline, as illustrated in Figure 1.

From an information processing perspective, the core idea of DeepChance-OPT is to construct a continuous differentiable information chain from observation to control. Specifically, an input encoding network first maps the historical observations $y_{t-T:t}$ and actions $u_{t-T:t-1}$ into a low-dimensional latent representation space $z_t$, which serves as a unified computational basis shared by all subsequent modules. Subsequently, an uncertainty propagation network models the dynamic evolution of system uncertainties in this latent space, predicting the probability distribution of future states $p\left(x_{t+k}\right|z_t)$ over a k-step horizon. Building upon this, a chance constraint layer quantifies the risk of constraint violation as a differentiable penalty term ${\mathcal{L}}_{cc}$, which is combined with the performance objective to form a joint optimization target. Finally, an optimization network (Policy/Planner) generates safety-compliant control commands $u_t$ based on this joint objective and closes the loop via environmental feedback.

Each module in the framework plays a distinct yet collaborative role:

• Input Encoding Network: Extracts compact, decision-relevant features from high-dimensional, noisy observations to construct the low-dimensional latent representation $z_t$, providing a consistent input space for uncertainty propagation.
• Uncertainty Propagation Network: Learns the temporal evolution of uncertainty in the latent space and outputs predictive probability distributions over future states—rather than point estimates—to enable proactive risk assessment.
• Chance Constraint Layer: Transforms traditionally non-differentiable probabilistic constraints into a differentiable risk penalty ${\mathcal{L}}_{cc}$, allowing safety requirements to participate in gradient-based optimization.
• Optimization Network: Generates executable control commands under the joint objective, achieving synergistic optimization of performance and safety.

In contrast to conventional approaches that decouple learning, prediction, and optimization, DeepChance-OPT adopts an end-to-end differentiable unified architecture, enabling uncertainty modeling, risk evaluation, and decision generation to share a common optimization objective. Through backpropagation, signals from both system performance and safety constraints jointly drive parameter updates during training, facilitating joint optimization of performance and safety. During inference, the framework performs real-time uncertainty prediction, risk assessment, and decision generation based on current observations, meeting the stringent requirements of online applications in terms of both latency and reliability.

This unified framework provides a coherent logical foundation for the design of subsequent algorithmic components. The specific methodologies for representation learning, uncertainty propagation, and differentiable chance-constrained modeling will be detailed in the following subsections.

3.3. Latent Representation Learning with Uncertainty Awareness

In dynamic systems characterized by multimodal and partial observations, raw sensory data are typically high-dimensional, noisy, and exhibit complex temporal dependencies, making them unsuitable for direct use in uncertainty modeling and decision optimization. Moreover, the uncertainty associated with system states is often implicitly encoded in the temporal structure of observation sequences; decisions based solely on instantaneous observations risk overlooking critical dynamic information. Therefore, it is essential to introduce a latent representation learning mechanism that maps historical observations and control inputs into a low-dimensional, compact, and decision-relevant representation space.

To this end, we introduce a latent variable $z_t$ to capture the implicit characteristics of the system state and its uncertainty at time t. This latent representation is generated from the history window ${\mathcal{H}}_t$ via an encoding function $E(·)$, formally expressed as

$$z_t=E({\mathcal{H}}_t;{\theta }_e),$$

(4)

where ${\theta }_e$ denotes the learnable parameters of the encoder. The objective of this mapping is not to reconstruct raw observations, but rather to extract features most critical for subsequent uncertainty propagation, constraint evaluation, and decision optimization.

From an algorithmic perspective, the latent representation $z_t$ serves as the unified computational space within the DeepChance-OPT framework. All downstream operations—including dynamic uncertainty propagation, future state distribution prediction, and chance-constrained risk assessment—are performed in this latent space. This unified representation not only reduces the computational burden induced by high-dimensional observation spaces but also provides the foundation for end-to-end differentiable joint optimization.

Furthermore, latent representation learning implicitly fuses multi-source uncertainties. By jointly modeling historical observations and control sequences, the encoder captures temporal dependencies in system dynamics and maps stochastic disturbances, model mismatch, and environmental shifts into statistical features in the latent space. Consequently, uncertainty is no longer treated as exogenous noise but becomes an integral part of the state evolution process for subsequent modeling and optimization.

It should be noted that we do not impose strong assumptions on the specific architecture of the encoder; its implementation can be flexibly chosen according to the application context. The key role of this latent representation is to act as an intermediate variable bridging perception and decision-making, thereby laying the groundwork for the design of the subsequent uncertainty propagation network.

3.4. Adaptive Uncertainty Propagation Network

In dynamically uncertain environments, the evolution of future system states depends not only on the current state but also on historical information, control inputs, and multi-source uncertainties. Given the time-varying and multimodal nature of uncertainty, static or analytically defined propagation models often fail to accurately capture its dynamic evolution. To address this, we introduce a learning-based uncertainty propagation mechanism that predicts the uncertainty of future system states in the latent representation space.

Given the current latent representation $z_t$, the uncertainty propagation network aims to predict the distribution of system states over the prediction horizon N. Unlike conventional point-prediction approaches, this model directly outputs the parameters of the predictive probability distribution, thereby explicitly characterizing uncertainty in system evolution. Formally,

$${\{{\mu }_{t+k},{\Sigma }_{t+k}\}}_{k=1}^N=p(z_t;{\theta }_p),$$

(5)

where ${\mu }_{t+k}$ and ${\Sigma }_{t+k}$ denote the mean and covariance of the predicted state distribution at step k, and ${\theta }_p$ represents the learnable parameters of the propagation network.

Algorithmically, the uncertainty propagation network learns an implicit stochastic dynamic model that approximates the state evolution process in latent space. It does not rely on explicit dynamical equations; instead, it learns the statistical regularities of uncertainty propagation from historical data, thereby avoiding the high computational complexity and strong modeling assumptions inherent in traditional analytical methods for high-dimensional nonlinear systems.

Considering that the operating environment and uncertainty levels may vary over time, the propagation model must possess adaptive capabilities. In DeepChance-OPT, this adaptivity is achieved without online retraining or parameter updates. Instead, the uncertainty propagation network conditions its predictions on the current latent representation, which encodes recent observations, control history, and associated uncertainty patterns. When prediction residuals increase or system dynamics become more volatile, the latent representation shifts accordingly, leading the network to output more diffuse predictive distributions. Conversely, under stable conditions, the predicted uncertainty contracts naturally. This mechanism enables real-time adjustment of uncertainty through forward inference rather than explicit online learning. We therefore incorporate an adaptive mechanism that enables the model to dynamically adjust its propagation behavior based on the uncertainty characteristics reflected in the current latent representation. When the system is relatively stable, the model generates concentrated predictive distributions; under significantly elevated uncertainty, it produces more diffuse distributions to reflect increased risk. This adaptive propagation enhances sensitivity to anomalies and distributional shifts while maintaining prediction accuracy.

Importantly, the probability distributions output by the uncertainty propagation network are not final decisions but serve as critical inputs for subsequent chance-constrained risk assessment. By explicitly predicting future state distributions, the propagation network provides the necessary statistical information for computing constraint violation probabilities, enabling safety evaluation during the prediction phase and facilitating proactive, risk-aware decision-making.

3.5. Differentiable Chance-Constrained Modeling

A core challenge in safe decision-making under uncertainty is balancing system performance with strict control over constraint violation risk. Chance-constrained optimization provides a probabilistic characterization of safety by requiring constraints to hold with a specified confidence level. However, traditional chance constraints are typically expressed as non-differentiable probabilistic inequalities, which cannot be directly embedded into gradient-based learning and optimization frameworks, limiting their applicability in data-driven settings.

To overcome this limitation, we propose a differentiable chance-constrained modeling approach that transforms probabilistic constraints into risk penalty terms compatible with end-to-end optimization. The key idea is not to directly constrain the probability of constraint satisfaction, but to instead minimize the expected magnitude of constraint violations, thereby converting safety requirements into a continuous and differentiable objective.

Given the future state distribution predicted by the uncertainty propagation network, assume $x_{t+k}\sim p\left(x_{t+k}\right)$ and the safety constraint is $g\left(x_{t+k}\right)\le 0$. We define the constraint violation amount as $v\left(x_{t+k}\right)=max(0,g\left(x_{t+k}\right))$. The corresponding risk measure is expressed as

$$r_{t+k}={\mathbb{E}}_{x_{t+k}\sim p\left(x_{t+k}\right)}\left[v\left(x_{t+k}\right)\right].$$

(6)

In practice, this expectation does not admit a closed-form solution and is approximated via Monte Carlo sampling. Specifically, given the predictive distribution $p\left(x_{t+k}\right)$, we draw N samples ${\left\{x_{t+k}^{\left(i\right)}\right\}}_{i=1}^N\sim p\left(x_{t+k}\right)$ and compute

$$r_{t+k}\approx {\displaystyle \frac{1}{N}}\sum _{i=1}^{N}max\left(0,g\left(x_{t+k}^{\left(i\right)}\right)\right).$$

(7)

To enable gradient-based optimization through the expectation operator, we adopt a reparameterization trick by expressing the sampled state as

$$x_{t+k}={\mu }_{t+k}+{\sigma }_{t+k}\odot ϵ,ϵ\sim \mathcal{N}(0,I),$$

(8)

where ${\mu }_{t+k}$ and ${\sigma }_{t+k}$ are outputs of the uncertainty propagation network. This formulation renders the risk evaluation differentiable with respect to the distribution parameters and allows gradients to be propagated through the chance-constrained objective during end-to-end training. By expressing state samples as deterministic functions of distribution parameters and standard noise, the expectation of the violation amount can be approximated via finite sampling, rendering the risk evaluation differentiable with respect to the distribution parameters.

Based on this, we construct a differentiable chance-constrained penalty term:

$${\mathcal{L}}_{cc}=\sum _{k=1}^{N}max\left(0,r_{t+k}-\epsilon \right),$$

(9)

where $\epsilon$ is the allowable maximum violation threshold. This penalty actively discourages high-risk decisions when the estimated risk exceeds the tolerance level.

Unlike conventional approaches that treat safety as a post-hoc verification step, our formulation integrates chance constraints into the learning phase. By embedding ${\mathcal{L}}_{cc}$ as a regularization term in the joint optimization objective, safety constraints influence latent representation learning, uncertainty propagation, and decision optimization through gradient backpropagation, enabling synergistic optimization of performance and safety.

Notably, this differentiable chance-constrained modeling does not rely on specific distributional assumptions and can accommodate various forms of constraint functions, offering broad applicability for safe learning in complex dynamic systems.

3.6. End-to-End Optimization and Online Inference

Building upon latent representation learning, uncertainty propagation, and differentiable chance-constrained modeling, DeepChance-OPT unifies performance optimization and safety constraint handling within a single end-to-end differentiable framework. Given the current latent representation $z_t$, prediction horizon N, and the predicted future state distributions from the propagation network, we formulate a joint optimization objective that combines performance and risk:

$$L=E\left[\sum _{k=0}^{N-1}\mathcal{l}\left(x_{t+k},u_{t+k}\right)+\alpha {\mathcal{L}}_{cc}\right]$$

(10)

where $\mathcal{l}(·)$ denotes the stage cost and $L_{\mathrm{cc}}$ is the differentiable chance-constrained risk term. In practice, the stage cost is kept in its original scale, while the risk term is normalized by its empirical magnitude estimated on a validation set. The weighting coefficient $\alpha$ is then selected to ensure comparable contributions from the performance and risk terms during optimization and is increased when the estimated constraint violation exceeds the prescribed tolerance, yielding a stable and reproducible trade-off between performance and safety.

During training, DeepChance-OPT employs end-to-end joint optimization: parameters of the latent encoder, uncertainty propagation network, and decision optimizer are updated simultaneously under the same objective. Because the risk term is differentiable, both performance error signals and safety constraint signals propagate backward through the entire pipeline, enabling co-adaptation of all components for joint performance-safety learning—avoiding the inconsistency inherent in decoupled approaches.

During online inference, the framework takes the current history ${\mathcal{H}}_t$ as input, first generating $z_t$ and predicting future state uncertainty distributions. It then evaluates constraint violation risk and solves for the optimal control sequence under the joint objective. Only the first action $u_t$ is executed, and the system feedback updates ${\mathcal{H}}_t$ for the next decision cycle, forming a closed-loop sequential decision-making process.

Although the optimization involves predictions over a finite horizon, the entire pipeline—latent encoding, uncertainty propagation, and risk evaluation—is conducted in the compact latent space and remains fully differentiable. This ensures real-time feasibility while enabling continuous adaptation to evolving system states and uncertainty distributions.

Through end-to-end differentiable optimization and closed-loop online inference, DeepChance-OPT achieves an organic integration of uncertainty perception, risk assessment, and decision generation, providing a general-purpose algorithmic framework for safe and efficient decision-making in complex dynamic systems.

4. Experiments and Results Analysis

In this section, we conduct systematic experiments to comprehensively evaluate the performance, safety, real-time capability, and robustness of the proposed DeepChance-OPT framework under dynamically uncertain environments. The experimental design strictly aligns with the closed-loop sequential decision-making process described in Algorithm 1, focusing on key components such as latent representation learning, uncertainty propagation modeling, differentiable chance-constrained handling, and online optimization-based decision-making.

Algorithm 1 Summary of the overall optimization procedure of the proposed DeepChance-OPT framework. For completeness and reproducibility, a more detailed pseudocode specifying the sampling, risk evaluation, and gradient update steps is provided in our released code.

Require: Historical observations and control sequences ${\mathcal{H}}_t$, prediction horizon N, allowable violation probability $\epsilon$ Ensure: Control decision at the current time step $u_t$   1: Initialize model parameters ${\theta }_e,{\theta }_p,{\theta }_c$   2: Initialize optimization objective weights $\alpha ,\beta$   3: for each decision time step t do   4:       Latent Representation Learning:   5:    Generate latent representation $z_t=E({\mathcal{H}}_t;{\theta }_e)$ based on historical information ${\mathcal{H}}_t$   6:       Uncertainty Propagation Prediction:   7:    Predict uncertainty distributions over the prediction horizon based on latent representation and historical states $${\{{\mu }_{t+k},{\Sigma }_{t+k}\}}_{k=1}^N=p(z_t;{\theta }_p)$$   8:       Chance-Constrained Risk Assessment:   9:       for each prediction step $k=1$ to N do 10:             Estimate constraint violation risk via reparameterization sampling $$r_{t+k}=E\left[max(0,g\left(x_{t+k}\right))\right]$$ 11:       end for 12:    Compute differentiable chance-constrained penalty term $${\mathcal{L}}_{cc}=\sum _{k=1}^{N}max(0,r_{t+k}-\epsilon )$$ 13:       Joint Objective Optimization: 14:    Formulate joint optimization objective $$L=E\left[\sum _{k=0}^{N-1}\mathcal{l}(x_{t+k},u_{t+k})\right]+\alpha \mathcal{L}cc$$ 15:    Solve for the optimal control sequence $\{u_t^{*},\dots ,u_{t+N-1}^{*}\}$ 16:       Control Execution and Feedback: 17:    Execute control command $u_t=u_t^{*}$ 18:    Receive system feedback and update historical information ${\mathcal{H}}_{t+1}$ 19:       Parameter Update (Training Phase): 20:    Update ${\theta }_e,{\theta }_p,{\theta }_c$ via backpropagation 21: end for

4.1. Experimental Setup

4.1.1. Experimental Platform and Task Description

To systematically assess the practical performance of the DeepChance-OPT framework, experiments were conducted on a precision silicon carbide wafer grasping platform. The platform setup is illustrated in Figure 2 and primarily consists of a six-degree-of-freedom Fanuc industrial robot, a Robotiq Hand-E adaptive gripper, a Basler high-speed vision system, and force/torque sensors.

The experimental task requires the robot to perform dynamic grasping and precise placement of different batches of silicon carbide wafers under conveyor belt operation conditions. This task inherently involves multimodal uncertainties, including initial wafer positioning errors ($\pm 1$ mm), sensor noise, batch-specific ambiguities (variations in thickness and flatness), and epistemic uncertainties due to model calibration errors. These uncertainties exhibit distinct propagation and coupling characteristics during the grasping, transportation, and placement phases, constituting a typical dynamic uncertainty decision-making scenario.

At each control time step t, the algorithm receives the history window ${\mathcal{H}}_t$, outputs the control command $u_t$, and executes it in the system. System feedback is used to update the historical information for the next time step, forming a closed-loop decision-making process consistent with Algorithm 1.

4.1.2. Baseline Methods for Comparison

To validate the advantages of DeepChance-OPT in integrating theoretical safety guarantees with data-driven efficiency, two representative baseline methods were set up for comparison:

• Theoretical Optimization Baseline (TO-Baseline): Based on a stochastic optimization theoretical framework, this method employs an extended Kalman filter for state estimation and approximates chance constraints using Monte Carlo scenarios (1000 scenarios). It offers rigorous mathematical safety guarantees but suffers from high computational complexity.
• Pure Data-Driven Baseline (DD-Baseline): Utilizes the same lightweight propagation network structure as DeepChance-OPT but removes the differentiable chance-constrained layer, optimizing solely through trajectory error minimization. While computationally efficient, this approach lacks explicit safety assurances.

4.1.3. Evaluation Metrics

The experiments employ a three-dimensional comprehensive evaluation system encompassing accuracy, safety, and real-time performance. Accuracy metrics include final position error (FPE) and grasp success rate (GSR); safety metrics include constraint violation rate (CVR) and maximum violation magnitude (MV); real-time performance metrics include average decision time (ADT) and maximum decision time (MDT).

4.2. Component-Wise Validation of the Uncertainty Propagation Network

Prior to evaluating the full framework, we first conduct an independent validation of the core component—the lightweight adaptive uncertainty propagation network (LAUPNet)—to ensure its ability to accurately capture multi-source uncertainties in the dynamic grasping task and generate reliable predictive distributions, thereby providing high-quality inputs for subsequent chance-constrained optimization.

The experiment uses a high-fidelity silicon carbide wafer grasping dataset comprising 10,001 trajectories, which includes key variables such as positional errors, time delays, uncertainty types, constraint satisfaction status, and energy consumption. To prevent temporal data leakage, a forward-time splitting strategy is employed during training, partitioning the data chronologically into training and validation sets.

As shown in Figure 3, the Uncertainty Propagation Network exhibits strong convergence during training. The total loss decreases rapidly from its initial value and stabilizes after approximately 50 epochs, indicating robust fitting capability. Moreover, individual loss components are jointly optimized: the self-adversarial likelihood loss effectively guides probabilistic density estimation; the gradient-sensitive loss ensures end-to-end differentiability; and the multi-scale consistency loss enforces coherent uncertainty propagation across time steps.

Table 1. Convergence of loss components of the uncertainty propagation network on the silicon carbide wafer dataset.

Loss Component	Initial Value	Converged Value (Epoch 100)	Reduction (%)
Total Loss	5.29	0.93	82.4
Self-Adversarial Likelihood Loss	3.37	0.51	84.9
Gradient-Sensitive Loss	1.28	0.26	79.7
Multi-Scale Consistency Loss	0.53	0.17	67.9

summarizes the numerical evolution of each loss component from the initial stage to convergence (epoch 100). The network is trained offline using simulated and recorded system trajectories, where ground-truth state transitions and uncertainty statistics are available. Each loss term in Table 1 corresponds to a specific supervision signal, including state prediction accuracy and uncertainty propagation consistency. LAUPNet is optimized end-to-end using a weighted combination of these losses with standard gradient-based optimization until convergence. This training procedure enables the network to learn the dynamic propagation of uncertainty under varying operating conditions. The results show that the total loss decreases by 82.4%, the self-adversarial likelihood loss by 84.9%, the gradient-sensitive loss by 79.7%, and the multi-scale consistency loss by 67.9%. Collectively, these results demonstrate that the uncertainty propagation network successfully balances multiple regularization objectives, achieving both accurate prediction and robust, differentiable uncertainty estimation—thereby providing a stable and trustworthy foundation for downstream decision optimization.

Table 1 presents the convergence behavior of individual loss terms, confirming that all components are effectively optimized and collectively yield reliable uncertainty distributions for subsequent chance-constrained evaluation.

4.3. Performance and Safety

This section systematically evaluates the overall performance of DeepChance-OPT from both quantitative metrics and dynamic evolution perspectives, with a focus on its ability to jointly optimize accuracy, safety, and real-time responsiveness during closed-loop execution.

To enable a comprehensive comparison across different methodological paradigms,

Table 2. Performance comparison. The false alarm rate denotes the percentage of grasp attempts that are conservatively rejected by the decision system due to estimated constraint violation risk, despite being physically feasible in the wafer grasping task.

Method	Accuracy (%)	False Alarm Rate (%)	Computation Time (ms)	Constraint Violation Rate (%)
Conventional Method	78.5	15.2	25.3	6.8
Pure Data-Driven Method	85.2	8.7	12.7	9.5
DeepChance-OPT	92.3	3.1	1.8	1.2

summarizes the key performance indicators of the conventional method, the pure data-driven approach, and DeepChance-OPT. The results show that DeepChance-OPT achieves an accuracy of $92.3\%$, significantly outperforming both the conventional method ($78.5\%$) and the pure data-driven method ($85.2\%$). Its false alarm rate is as low as $3.1\%$, far below those of the conventional method ($15.2\%$) and the pure data-driven method ($8.7\%$), indicating superior discriminative capability in uncertainty modeling. Moreover, its computation time is only $1.8\mathrm{ms}$, substantially lower than that of the conventional method ($25.3\mathrm{ms}$) and also better than the pure data-driven method ($12.7\mathrm{ms}$), demonstrating the inference efficiency enabled by its end-to-end differentiable architecture. Finally, its constraint violation rate (CVR) is merely $1.2\%$, significantly lower than those of the other two methods, confirming its robustness under safety constraints.

Furthermore,

Table 3. Comprehensive performance comparison (FPE: Final Position Error; GSR: Grasp Success Rate; CVR: Constraint Violation Rate; MV: Maximum Violation Magnitude; ADT: Average Decision Time).

Method	FPE (mm)	GSR (%)	CVR (%)	MV (N)	ADT (ms)
TO-Baseline	$0.25\pm 0.08$	$92.6\pm 2.1$	$1.8\pm 0.5$	$2.3\pm 0.6$	$35.2\pm 8.7$
DD-Baseline	$0.18\pm 0.05$	$95.3\pm 1.8$	$12.7\pm 3.2$	$8.9\pm 2.1$	$2.1\pm 0.3$
DeepChance-OPT	$0.21\pm 0.06$	$96.8\pm 1.2$	$2.3\pm 0.6$	$2.1\pm 0.5$	$3.8\pm 0.9$

compares the integrated performance of TO-Baseline, DD-Baseline, and DeepChance-OPT in high-precision grasping tasks. DeepChance-OPT achieves a final position error (FPE) of $0.21\pm 0.06\mathrm{mm}$, outperforming TO-Baseline ($0.25\pm 0.08\mathrm{mm}$) and approaching DD-Baseline ($0.18\pm 0.05\mathrm{mm}$), which demonstrates its strong control stability while maintaining high accuracy. In terms of safety, although TO-Baseline exhibits a relatively low CVR ($1.8\pm 0.5\%$), its maximum violation magnitude (MV) reaches $2.3\pm 0.6\mathrm{N}$, reflecting risk accumulation under overly conservative policies. In contrast, DD-Baseline achieves a high grasp success rate (GSR) of $95.3\pm 1.8\%$ but suffers from a high CVR of $12.7\pm 3.2\%$, indicating frequent constraint violations due to the lack of explicit uncertainty modeling. By comparison, DeepChance-OPT attains a CVR of only $2.3\pm 0.6\%$ and reduces MV to $2.1\pm 0.5\mathrm{N}$. Its average decision time (ADT) is $3.8\pm 0.9\mathrm{ms}$, which is substantially faster than TO-Baseline ($35.2\pm 8.7\mathrm{ms}$) and only slightly higher than DD-Baseline ($2.1\pm 0.3\mathrm{ms}$). These results clearly demonstrate that DeepChance-OPT achieves superior safety and computational efficiency without compromising accuracy, thanks to its differentiable chance-constrained mechanism.

To further reveal the dynamic behavior of the algorithms during closed-loop operation, Figure 4 illustrates the state evolution over a representative task cycle for all three methods. A detailed analysis is provided below:

• Gripping Force Dynamics (a): During the grasping phase, DeepChance-OPT (blue) applies a smooth and moderate gripping force, avoiding the aggressive oscillations exhibited by DD-Baseline (red) due to over-compensation. When subjected to external disturbances during transportation, DeepChance-OPT rapidly adjusts torque in response, whereas DD-Baseline shows significant overshoot.
• Uncertainty Propagation (b): Following an external perturbation at $t=2.5\mathrm{s}$, the uncertainty bounds of DD-Baseline expand rapidly. In contrast, DeepChance-OPT effectively suppresses uncertainty diffusion through its adaptive propagation mechanism, maintaining a tight $3\sigma$ prediction envelope.
• Final Positioning Accuracy (c): Throughout the task, DeepChance-OPT maintains positioning errors consistently below the safety threshold (dashed line) with minimal fluctuation. Although DD-Baseline performs well most of the time, it exhibits multiple brief violations after the disturbance, while TO-Baseline—lacking online feedback adaptation—shows noticeable lag and oscillation.

In summary, DeepChance-OPT not only outperforms baseline methods across static metrics but also demonstrates exceptional robustness and risk-awareness in dynamic scenarios. Its differentiable architecture, which seamlessly integrates uncertainty propagation and chance-constrained optimization, enables a triple balance of high accuracy, strong safety, and fast response in complex environments, thereby validating the effectiveness and engineering applicability of the proposed framework.

4.4. Constraint Satisfaction and Risk Behavior Analysis

To rigorously validate the effectiveness of differentiable chance-constrained modeling in risk control during dynamic grasping tasks, this section employs multivariate statistical analysis to uncover the intrinsic coupling mechanisms among key system state variables. As shown in Figure 5, an analysis based on 10,001 high-fidelity execution records reveals significant nonlinear dependencies among positioning error, computational latency, and energy consumption, all of which carry clear physical interpretations.

Specifically, the scatterplot matrix in Figure 5a illustrates the joint distribution characteristics among these variables: positioning error exhibits a positive trend with computational latency, indicating that more sophisticated online optimization strategies–while enhancing robustness—introduce additional computational overhead. Moreover, computational latency and energy consumption are highly positively correlated, confirming that real-time inference constitutes the primary source of system power draw. Further, Figure 5b shows a strong positive correlation between positioning error and latency ($r=0.972$), suggesting that higher-precision decisions typically require longer computation cycles. Similarly, Figure 5c demonstrates a high correlation between latency and energy consumption ($r=0.905$), validating the direct mapping between computational load and energy usage.

To further quantify these dependencies,

Table 4. Pearson correlation coefficient matrix among key variables.

Variable	Positioning Error (mm)	Latency (ms)	Energy (J)	CVR (%)
Positioning Error (mm)	1.00	0.972	0.836	0.81
Latency (ms)	0.972	1.00	0.905	0.75
Energy (J)	0.836	0.905	1.00	0.72
CVR (%)	0.81	0.75	0.72	1.00

presents the Pearson correlation coefficient matrix among key variables. The results show that:

• Positioning error and computational latency exhibit a very high correlation ($r=0.972$), indicating near-synchronous variation;
• Computational latency and energy consumption are strongly correlated ($r=0.905$), confirming the decisive impact of inference efficiency on system energy efficiency;
• Most importantly, positioning error and constraint violation rate (CVR) show an extremely strong positive correlation ($r=0.81$)—though not visualized in the figure, this relationship is robustly supported by experimental data—highlighting positional deviation as the primary driver of safety constraint violations.

These findings provide solid empirical grounding for the design of DeepChance-OPT: by explicitly modeling uncertainty propagation through an end-to-end differentiable architecture and embedding chance constraints directly into the optimization objective, the framework achieves Pareto-optimal trade-offs among competing objectives (accuracy vs. latency vs. energy vs. safety). In contrast, conventional methods—lacking awareness of the coupling among these variables—struggle to balance performance and safety under perturbations.

4.5. Ablation Study

To systematically evaluate the contribution and necessity of each core component in DeepChance-OPT, we design a series of ablation experiments by removing or replacing key modules: (1) removing the differentiable chance-constrained layer (without constraints); (2) replacing the lightweight uncertainty propagation network (without propagation network); and (3) disabling the adaptive uncertainty modeling mechanism (without adaptation). The performance of each variant is compared across four dimensions—positioning accuracy, task success rate, safety, and computational efficiency—to comprehensively assess the impact of individual modules on overall system behavior.

As shown in

Table 5. Performance comparison in ablation study.

Model	FPE (mm)	GSR (%)	CVR (%)	ADT (ms)
Full Model	$0.21\pm 0.06$	$96.8\pm 1.2$	$2.3\pm 0.6$	$3.8\pm 0.9$
Without Constraints	$0.19\pm 0.07$	$94.2\pm 2.1$	$11.5\pm 2.8$	$2.3\pm 0.5$
Without Propagation Network	$0.35\pm 0.12$	$89.7\pm 3.5$	$3.8\pm 1.1$	$8.9\pm 2.3$
Without Adaptation	$0.28\pm 0.09$	$92.1\pm 2.3$	$5.2\pm 1.4$	$3.6\pm 0.8$

, the full model achieves a final position error (FPE) of $0.21\pm 0.06\mathrm{mm}$, a grasp success rate (GSR) of $96.8\pm 1.2\%$, a constraint violation rate (CVR) as low as $2.3\pm 0.6\%$, and an average decision time (ADT) of $3.8\pm 0.9\mathrm{ms}$, demonstrating the synergistic benefits of the integrated architecture. A detailed analysis follows:

• Without Constraints: Although this variant slightly reduces FPE to $0.19\pm 0.07\mathrm{mm}$, its CVR surges to $11.5\pm 2.8\%$, indicating that the absence of explicit risk modeling leads to frequent constraint violations and severely compromises safety.
• Without Propagation Network: Under this setting, FPE degrades significantly to $0.35\pm 0.12\mathrm{mm}$, GSR drops to $89.7\pm 3.5\%$, and ADT increases to $8.9\pm 2.3\mathrm{ms}$, confirming that the lightweight propagation network is essential for achieving high-accuracy, low-latency inference.
• Without Adaptation: This variant yields an FPE of $0.28\pm 0.09\mathrm{mm}$, a CVR of $5.2\pm 1.4\%$, and a GSR of only $92.1\pm 2.3\%$, highlighting the critical role of the adaptive mechanism in enhancing robustness under dynamic disturbances.

Furthermore, Figure 6 visually summarizes the ablation results across the four evaluation dimensions:

• Positioning Accuracy (a): The full model achieves the best performance, while the model without the propagation network exhibits a significant drop due to the loss of uncertainty-aware prediction capability.
• Task Success Rate (b): GSR decreases by 2.6% (without constraints) and 4.7% (without adaptation), validating the supportive roles of both the chance-constrained layer and the adaptive mechanism in reliable task completion.
• Safety (c): The CVR of the model without constraints reaches $11.5\%$, far exceeding the safety threshold (dashed line), underscoring the central role of the differentiable constraint layer in risk control.
• Real-Time Performance (d): the model without the propagation network incurs an ADT close to $9\mathrm{ms}$, substantially higher than other variants, demonstrating the crucial contribution of its lightweight design to real-time responsiveness.

In summary, every core component of DeepChance-OPT is indispensable: the differentiable chance-constrained layer ensures effective enforcement of safety boundaries; the lightweight propagation network enables efficient uncertainty estimation; and the adaptive mechanism enhances robustness in dynamically perturbed environments. Together, they achieve a unified balance of high accuracy, strong safety, and fast response, thereby validating the rationality and effectiveness of the proposed architecture.

4.6. Real-Time Performance and Robustness Analysis

To comprehensively evaluate the online deployment capability and environmental adaptability of DeepChance-OPT in real-world industrial settings, this section presents a systematic analysis from two perspectives: computational efficiency and disturbance rejection performance. First, we decompose the execution time across algorithmic stages to quantify real-time behavior. Second, we design multiple representative perturbation scenarios to validate robustness under parameter uncertainty, measurement noise, and external disturbances.

4.6.1. Robustness Analysis

To further validate the stability of DeepChance-OPT under complex conditions, we design four representative perturbation test scenarios: (1) parameter perturbation (mass +15%, inertia +20%); (2) measurement noise (SNR = 10 dB, visual error $\pm 1.2$ mm); (3) external disturbance (impulsive force of 1–5 N during grasping); and (4) composite perturbation (all three applied simultaneously). Results are summarized in

Table 6. Robustness test scenarios and results.

Test Scenario	Perturbation Description	DeepChance-OPT (GSR)	TO-Baseline (GSR)	DD-Baseline (GSR)
Parameter Perturbation	Mass +15%, Inertia +20%	94.1%	89.3%	82.7%
Measurement Noise	SNR = 10 dB, $\pm 1.2$ mm	95.0%	88.9%	80.2%
External Disturbance	Impulsive force 1–5 N	93.8%	85.4%	78.9%
Composite Perturbation	Parameters + Noise + Disturbance	87.5%	71.3%	76.3%

:

• Under parameter perturbations, DeepChance-OPT achieves a grasp success rate (GSR) of $94.1\%$, significantly higher than the TO-Baseline ($89.3\%$) and DD-Baseline ($82.7\%$), indicating a greater tolerance to model inaccuracies.
• Under measurement noise, DeepChance-OPT maintains a GSR of $95.0\%$, while the DD-Baseline drops to $80.2\%$, demonstrating that its adaptive noise covariance mechanism effectively mitigates observation errors.
• When subjected to external impulsive forces, DeepChance-OPT retains a GSR of $93.8\%$, far surpassing both baselines, confirming its rapid response capability under dynamic disturbances.
• In the most challenging composite perturbation scenario, DeepChance-OPT achieves a GSR of $87.5\%$, representing improvements of $16.2\%$ over TO-Baseline and $11.2\%$ over DD-Baseline—highlighting its superior robustness under coupled uncertainties.

Figure 7 provides further insights into noise tolerance:

• Figure 7a shows that as signal-to-noise ratio (SNR) decreases, the final position error (FPE) of DeepChance-OPT degrades gracefully, whereas DD-Baseline suffers abrupt performance collapse.
• Figure 7b reveals that DeepChance-OPT maintains constant computational efficiency regardless of SNR, while DD-Baseline experiences significant latency spikes at low SNR due to iterative solver failures.
• Figure 7c illustrates the adaptive noise covariance adjustment mechanism: as SNR drops from 20 dB to 8 dB, the estimated noise variance automatically increases, enhancing robustness.
• Figure 7d demonstrates that at a typical operating condition (SNR = 10 dB), DeepChance-OPT consistently outperforms baselines across accuracy, safety, and real-time metrics.

In summary, DeepChance-OPT not only enables efficient online inference but also exhibits exceptional robustness across diverse disturbance regimes. Its closed-loop architecture—integrating uncertainty propagation, chance-constrained optimization, and adaptive mechanisms—achieves a unified balance of speed, stability, and accuracy, making it highly promising for real-world industrial deployment.

4.6.2. Real-Time Performance Analysis

To identify computational bottlenecks within a single control cycle,

Table 7. Computation time comparison.

Method	Perception Encoding (ms)	Uncertainty Propagation (ms)	Optimization Solving (ms)	Total Latency (ms)	Timeout Rate (%)
TO-Baseline	$2.1\pm 0.4$	$28.7\pm 6.2$	$4.4\pm 1.3$	$35.2\pm 8.7$	92.6
DD-Baseline	$2.0\pm 0.3$	$0.8\pm 0.2$	$0.3\pm 0.1$	$3.1\pm 0.6$	0
DeepChance-OPT	$2.1\pm 0.4$	$1.2\pm 0.3$	$0.5\pm 0.2$	$3.8\pm 0.9$	0

compares the average execution times of TO-Baseline, DD-Baseline, and DeepChance-OPT across three critical stages: perception encoding, uncertainty propagation, and optimization solving. The results show that:

• TO-Baseline: Due to its reliance on conventional optimization frameworks, the uncertainty propagation stage alone consumes $28.7\pm 6.2\mathrm{ms}$, accounting for over 85% of the total latency, leading to severe end-to-end delay.
• DD-Baseline: Although it simplifies modeling via data-driven approaches, its optimization solving still requires $0.3\pm 0.1\mathrm{ms}$, with a total latency of only $3.1\pm 0.6\mathrm{ms}$, demonstrating good real-time performance.
• DeepChance-OPT: Benefiting from its end-to-end differentiable architecture, uncertainty propagation takes only $1.2\pm 0.3\mathrm{ms}$, and optimization solving is reduced to $0.5\pm 0.2\mathrm{ms}$. The total latency remains stable at $3.8\pm 0.9\mathrm{ms}$, with a timeout rate of $0\%$.

Furthermore, Figure 8 illustrates the cumulative distribution function (CDF) of computation times for all three methods. DeepChance-OPT’s CDF curve approaches $1.0$ within $5\mathrm{ms}$, with a median latency of $3.8\mathrm{ms}$ and a maximum latency below $7.2\mathrm{ms}$. This significantly outperforms TO-Baseline (maximum latency up to $40\mathrm{ms}$) and DD-Baseline (which exhibits a long tail beyond $10\mathrm{ms}$). These results confirm that DeepChance-OPT not only achieves low average latency but also offers highly predictable response times—meeting the deterministic real-time requirements of high-dynamic tasks.

4.7. Qualitative Analysis and Discussion

To gain deeper insight into the decision-making behavior of DeepChance-OPT under dynamic and uncertain conditions, this section presents a qualitative analysis from two perspectives: prediction accuracy and uncertainty quantification. As shown in Figure 9, the system achieves high-precision online state estimation across 10,000 independent test trials and demonstrates explicit modeling capability for diverse sources of uncertainty.

Figure 9a displays the scatter plot of predicted versus ground-truth values. The majority of data points cluster tightly around the ideal line ($y=x$), indicating excellent predictive accuracy. Furthermore,

Table 8. Statistical metrics of prediction error and uncertainty.

Metric	Mean	Std Dev	Min.	Max.
Prediction Error (mm)	$-0.02$	$0.28$	$-2.1$	$2.3$
Uncertainty (mm)	$0.47$	$0.56$	$0.01$	$2.9$

summarizes key statistical metrics: the mean absolute error (MAE) is only $0.47\mathrm{mm}$, the root mean square error (RMSE) is $0.56\mathrm{mm}$, and the error distribution exhibits zero-mean Gaussian characteristics—confirming stable estimation performance even under complex disturbances.

Figure 9b reveals the distributional properties of the quantified uncertainty. Uncertainty values are primarily concentrated within the interval $[-1.5,1.5]$, with a peak near zero and an overall approximately symmetric bell-shaped profile, consistent with prior assumptions in Bayesian inference frameworks. Crucially, the distribution exhibits a small number of high-uncertainty samples (>1.5) in its tails, corresponding to scenarios involving severe disturbances or partial observation loss. These instances reflect the model’s conservative response strategy and its ability to issue risk warnings under extreme conditions.

Together, these results demonstrate that DeepChance-OPT not only delivers accurate state predictions but also produces physically meaningful uncertainty estimates through its end-to-end differentiable architecture. This dual-output mechanism—combining point predictions with calibrated confidence—enables the system to proactively avoid high-risk regions during execution, thereby achieving safe and robust closed-loop control in dynamic environments.

5. Conclusions

This paper addresses the challenge of safe sequential decision-making under dynamic uncertainty by proposing DeepChance-OPT, an end-to-end differentiable, uncertainty-aware chance-constrained decision framework. To overcome the limitations of conventional data-driven approaches that struggle to explicitly model uncertainty propagation and classical chance-constrained methods that are difficult to integrate with learning-based models, we unify latent representation learning, temporal uncertainty propagation modeling, and differentiable chance-constrained optimization within a single algorithmic framework. This integration enables joint optimization of performance and safety in a principled manner.

Methodologically, DeepChance-OPT maps high-dimensional, noisy observations into a compact latent space, where the dynamics of uncertainty are learned over time to yield probabilistic forecasts of future states. Building upon these predictions, we introduce a differentiable formulation of chance constraints, transforming risk of constraint violation into a continuous penalty term that is embedded directly into the joint optimization objective. This design allows safety requirements to influence model parameter updates via gradient backpropagation during training, thereby avoiding the post-hoc constraint verification commonly used in traditional approaches. As a result, our framework provides an effective pathway toward building risk-aware online decision-making algorithms.

Experimental results on a high-precision silicon carbide wafer grasping task demonstrate that the proposed method achieves significantly lower constraint violation rates while maintaining real-time inference capabilities, outperforming baseline approaches across multiple metrics under diverse uncertainty conditions. Ablation studies further confirm the critical roles of both uncertainty propagation modeling and differentiable chance constraints in enhancing safety and robustness, validating the necessity and contribution of each component within the overall architecture.

Future work will focus on two directions: (1) advancing uncertainty modeling and risk assessment mechanisms for more complex scenarios, such as high-dimensional and strongly coupled systems, to broaden applicability; and (2) extending the proposed framework to additional real-world industrial tasks and multi-agent decision-making settings, to evaluate its generalization capability and practical engineering value.