Search Results (195)

This work establishes definitive conditions for the inheritance of categorical completeness and cocompleteness by categories of internal group objects. We prove that while the completeness of $\mathbf{Grp}\left(\mathcal{C}\right)$ follows unconditionally from the completeness of the base category $\mathcal{C}$, cocompleteness requires $\mathcal{C}$ to be regular, cocomplete, and admit a free group functor left adjoint to the forgetful functor. Explicit limit and colimit constructions are provided, with colimits realized via coequalizers of relations induced by group axioms over free group objects. Applications demonstrate cocompleteness in topological groups, ordered groups, and group sheaves, while Lie groups serve as counterexamples revealing necessary analytic constraints—particularly the impossibility of equipping free groups on non-discrete manifolds with smooth structures. Further results include the inheritance of regularity when the free group functor preserves finite products, the existence of internal hom-objects in locally Cartesian closed settings, monadicity for locally presentable $\mathcal{C}$, and homotopical extensions where model structures on $\mathbf{Grp}\left(\mathcal{M}\right)$ reflect those of $\mathcal{M}$. This framework unifies classical category theory with geometric obstruction theory, resolving fundamental questions on exactness transfer and enabling new constructions in homotopical algebra and internal representation theory. Full article

Gas-phase chemistry has been identified as a major computational bottleneck in both the forward and adjoint modes of chemical transport models (CTMs). Although previous studies have demonstrated the potential of deep learning models to simulate and accelerate this process, few studies have examined the applicability and performance of these models in adjoint sensitivity analysis. In this study, a deep learning emulator for gas-phase chemistry is developed and trained on a diverse set of forward-mode simulations from the Community Multiscale Air Quality (CMAQ) model. The emulator employs a residual neural network (ResNet) architecture referred to as FiLM-ResNet, which integrates Feature-wise Linear Modulation (FiLM) layers to explicitly account for photochemical and non-photochemical conditions. Validation within a single timestep indicates that the emulator accurately predicts concentration changes for 74% of gas-phase species with coefficient of determination (R²) exceeding 0.999. After embedding the emulator into the CTM, multi-timestep simulation over one week shows close agreement with the numerical model. For the adjoint mode, we compute the sensitivities of ozone (O₃) with respect to O₃, nitric oxide (NO), nitrogen dioxide (NO₂), hydroxyl radical (OH) and isoprene (ISOP) using automatic differentiation, with the emulator-based adjoint results achieving a maximum R² of 0.995 in single timestep evaluations compared to the numerical adjoint sensitivities. A 24 h adjoint simulation reveals that the emulator maintains spatially consistent adjoint sensitivity distributions compared to the numerical model across most grid cells. In terms of computational efficiency, the emulator achieves speed-ups of 80×–130× in the forward mode and 45×–102× in the adjoint mode, depending on whether inference is executed on Central Processing Unit (CPU) or Graphics Processing Unit (GPU). These findings demonstrate that, once the emulator is accurately trained to reproduce forward-mode gas-phase chemistry, it can be effectively applied in adjoint sensitivity analysis. This approach offers a promising alternative approach to numerical adjoint frameworks in CTMs. Full article

The aerodynamic optimization of airfoil shapes remains a critical research area for enhancing aircraft performance under various flight conditions. In this study, the RAE 2822 airfoil was selected as a benchmark case to investigate and compare the effectiveness of surrogate-based methods under an Efficient Global Optimization (EGO) framework and an adjoint-based approach in both single-point and multi-point optimization settings. Prior to optimization, the computational fluid dynamics (CFD) model was validated against experimental data to ensure accuracy. For the surrogate-based methods, Kriging (KRG), Kriging with Partial Least Squares (KPLS), Gradient-Enhanced Kriging (GEK), and Gradient-Enhanced Kriging with Partial Least Squares (GEKPLS) were employed. In the single-point optimization, the GEK method achieved the highest drag reduction, outperforming other approaches, while in the multi-point case, GEKPLS provided the best overall improvement. Detailed comparisons were made against existing literature results, with the proposed methods showing competitive and superior performance, particularly in viscous, transonic conditions. The results underline the importance of incorporating gradient information into surrogate models for achieving high-fidelity aerodynamic optimizations. The study demonstrates that surrogate-based methods, especially those enriched with gradient information, can effectively match or exceed the performance of gradient-based adjoint methods within reasonable computational costs. Full article

This work illustrates the application of the “First-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integro-Differential Equations of Fredholm-Type” (1st-FASAM-NIDE-F) and the “Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integro-Differential Equations of Fredholm-Type” (2nd-FASAM-NIDE-F) to a paradigm heat transfer model. This physically based heat transfer model has been deliberately constructed so that it can be represented either by a neural integro-differential equation of a Fredholm type (NIDE-F) or by a conventional second-order “neural ordinary differential equation (NODE)” while admitting exact closed-form solutions/expressions for all quantities of interest, including state functions and first-order and second-order sensitivities. This heat transfer model enables a detailed comparison of the 1st- and 2nd-FASAM-NIDE-F versus the recently developed 1st- and 2nd-FASAM-NODE methodologies, highlighting the considerations underlying the optimal choice for cases where the neural net of interest is amenable to using either of these methodologies for its sensitivity analysis. It is shown that the 1st-FASAM-NIDE-F methodology enables the most efficient computation of exactly determined first-order sensitivities of the decoder response with respect to the optimized NIDE-F parameters, requiring a single “large-scale” computation for solving the 1st-Level Adjoint Sensitivity System (1st-LASS), regardless of the number of weights/parameters underlying the NIDE-F decoder, hidden layers, and encoder. The 2nd-FASAM-NIDE-F methodology enables the computation, with unparalleled efficiency, of the second-order sensitivities of decoder responses with respect to the optimized/trained weights. Full article

This work presents the “First-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integro-Differential Equations of Fredholm-Type” (1st-FASAM-NIDE-F) and the “Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integro-Differential Equations of Fredholm-Type” (2nd-FASAM-NIDE-F). It is shown that the 1st-FASAM-NIDE-F methodology enables the most efficient computation of exactly-determined first-order sensitivities of decoder response with respect to the optimized NIDE-F parameters, requiring a single “large-scale” computation for solving the 1st-Level Adjoint Sensitivity System (1st-LASS), regardless of the number of weights/parameters underlying the NIDE-F decoder, hidden layers, and encoder. The 2nd-FASAM-NIDE-F methodology enables the computation, with unparalleled efficiency, of the second-order sensitivities of decoder responses with respect to the optimized/trained weights, requiring only as many large-scale computations for solving the 2nd-Level Adjoint Sensitivity System (2nd-LASS) as there are non-zero feature functions of parameters. The application of both the 1st-FASAM-NIDE-F and the 2nd-FASAM-NIDE-F methodologies is illustrated in an accompanying work (Part II) by considering a paradigm heat transfer model. Full article

This work presents the mathematical frameworks of the “First-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integro-Differential Equations of Volterra-Type” (1st-FASAM-NIDE-V) and the “Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integro-Differential Equations of Volterra-Type” (2nd-FASAM-NIDE-V). It is shown that the 1st-FASAM-NIDE-V methodology enables the efficient computation of exactly-determined first-order sensitivities of the decoder response with respect to the optimized NIDE-V parameters, requiring a single “large-scale” computation for solving the 1st-Level Adjoint Sensitivity System (1st-LASS), regardless of the number of weights/parameters underlying the NIE-net. The 2nd-FASAM-NIDE-V methodology enables the computation, with unparalleled efficiency, of the second-order sensitivities of decoder responses with respect to the optimized/trained weights involved in the NIDE-V’s decoder, hidden layers, and encoder, requiring only as many “large-scale” computations as there are non-zero first-order sensitivities with respect to the feature functions. These characteristics of the 1st-FASAM-NIDE-V and 2nd-FASAM-NIDE-V are illustrated by considering a nonlinear heat conduction model that admits analytical solutions, enabling the exact verification of the expressions obtained for the first- and second-order sensitivities of NIDE-V decoder responses with respect to the model’s functions of parameters (weights) that characterize the heat conduction model. Full article

In this study, we analyze a time-fractional spatiotemporal $SIR$ model in a specific area $\Omega$. Taking into account the available resources, vaccines are allocated to region ${\omega }_1\subseteq \Omega$ and treatments to region ${\omega }_2\subseteq \Omega$, which may or may not coincide. Our objective is to minimize infections and costs by implementing an optimal regional control strategy. We establish the existence of optimal controls and related solutions, providing a characterization of optimal control in terms of state and adjoint functions. We employ the forward–backward sweep method to solve the associated optimality system numerically. The findings indicate that a combined strategy of vaccination and treatment is more effective in reducing disease transmission from adjacent regions. Full article

This study applied an adjoint data assimilation model capable of integrating wind fields to investigate a temperate storm surge event in the Bohai Sea region during October 18 to 21, 2024. Based on in situ water level measurements from five tide gauge stations, the model simulated the spatial distributions of water levels under different wind stress drag coefficients (C_D) schemes driven by reanalysis wind fields and interpolated wind fields. The results demonstrated that the scheme without the adjoint data assimilation exhibited relatively low accuracy. Upon integrating the adjoint data assimlation method, the errors of the reanalysis wind fields were reduced by 44%, while those of the interpolated wind fields experienced a 74% decrease in error magnitude. Overall, this study provides a reference for enhancing the accuracy of water level predictions during storm surge events. Full article

In this work, we present a numerical approach for solving optimal control problems for fluid heat transfer applications with a mixed optimality system: an FEM code to solve the adjoint solution over a precise restricted admissible solution set and an open-source well-known code for solving the state problem defined over a different one. In this way, we are able to decouple the optimality system and use well-established and validated numerical tools for the physical modeling. Specifically, two different CFD codes, OpenFOAM (finite volume-based) and FEMuS (finite element-based), have been used to solve the optimality system, while the data transfer between them is managed by the external library MEDCOUPLING. The state equations are solved in the finite volume code, while the adjoint and the control are solved in the finite element code. Two examples taken from the literature are implemented in order to validate the numerical algorithm: the first one considers a natural convection cavity resulting from a Rayleigh–Bénard configuration, and the second one is a conjugate heat transfer problem between a fluid and a solid region. Full article

Neural network (NN)-based controllers have emerged as a paradigm-shifting approach in modern control systems, demonstrating unparalleled capabilities in governing nonlinear dynamical systems with inherent uncertainties. This comprehensive review systematically investigates the theoretical foundations and practical implementations of NN controllers through the prism of Lyapunov stability theory, NN controller frameworks, and robustness analysis. The review establishes that recurrent neural architectures inherently address time-delayed state compensation and disturbance rejection, achieving superior trajectory tracking performance compared to classical control strategies. By integrating imitation learning with barrier certificate constraints, the proposed methodology ensures provable closed-loop stability while maintaining safety-critical operation bounds. Experimental evaluations using chaotic system benchmarks confirm the exceptional modeling capacity of NN controllers in capturing complex dynamical behaviors, complemented by formal verification advances through reachability analysis techniques. Practical demonstrations in aerial robotics and intelligent transportation systems highlight the efficacy of controllers in real-world scenarios involving environmental uncertainties and multi-agent interactions. The theoretical framework synergizes data-driven learning with nonlinear control principles, introducing hybrid automata formulations for transient response analysis and adjoint sensitivity methods for network optimization. These innovations position NN controllers as a transformative technology in control engineering, offering fundamental advances in stability-guaranteed learning and topology optimization. Future research directions will emphasize the integration of physics-informed neural operators for distributed control systems and event-triggered implementations for resource-constrained applications, paving the way for next-generation intelligent control architectures. Full article

Causal discovery involves searching intractably large spaces. Decomposing the search space into classes of observationally equivalent causal models is a well-studied avenue to making discovery tractable. This paper studies the topological structure underlying causal equivalence to develop a categorical formulation of Chickering’s transformational characterization of Bayesian networks. A homotopic generalization of the Meek–Chickering theorem on the connectivity structure within causal equivalence classes and a topological representation of Greedy Equivalence Search (GES) that moves from one equivalence class of models to the next are described. Specifically, this work defines causal models as propable symmetric monoidal categories (cPROPs), which define a functor category ${\mathcal{C}}^P$ from a coalgebraic PROP P to a symmetric monoidal category $\mathcal{C}$. Such functor categories were first studied by Fox, who showed that they define the right adjoint of the inclusion of Cartesian categories in the larger category of all symmetric monoidal categories. cPROPs are an algebraic theory in the sense of Lawvere. cPROPs are related to previous categorical causal models, such as Markov categories and affine CDU categories, which can be viewed as defined by cPROP maps specifying the semantics of comonoidal structures corresponding to the “copy-delete” mechanisms. This work characterizes Pearl’s structural causal models (SCMs) in terms of Cartesian cPROPs, where the morphisms that define the endogenous variables are purely deterministic. A higher algebraic K-theory of causality is developed by studying the classifying spaces of observationally equivalent causal cPROP models by constructing their simplicial realization through the nerve functor. It is shown that Meek–Chickering causal DAG equivalence generalizes to induce a homotopic equivalence across observationally equivalent cPROP functors. A homotopic generalization of the Meek–Chickering theorem is presented, where covered edge reversals connecting equivalent DAGs induce natural transformations between homotopically equivalent cPROP functors and correspond to an equivalence structure on the corresponding string diagrams. The Grothendieck group completion of cPROP causal models is defined using the Grayson–Quillen construction and relate the classifying space of cPROP causal equivalence classes to classifying spaces of an induced groupoid. A real-world domain modeling genetic mutations in cancer is used to illustrate the framework in this paper. Full article

For topology optimization problems under harmonic excitation in a frequency band, a large number of displacement and adjoint displacement vectors for different frequencies need to be computed. This leads to an unbearable computational cost, especially for large-scale problems. An effective approach, the Second-Order Arnoldi (SOAR) method, effectively solves the response and adjoint equations by projecting the original model to a reduced order model. The SOAR method generalizes the well-known Krylov subspace in a specified frequency point and can give accurate solutions for the frequencies near the specified point by using only a few basis vectors. However, for a wide frequency band, more expansion points are needed to obtain the required accuracy. This brings up the question of how many points are needed for an arbitrary frequency band. The traditional reduced order method improves the accuracy by uniformly increasing the expansion points. However, this leads to the redundancy of expansion points, as some frequency bands require more expansion points while others only need a few. In this paper, a bisection-based adaptive SOAR method (ASOAR), in which the points are added adaptively based on a local error estimation function, is developed to solve this problem. In this way, the optimal number and position of expansion points are adaptively determined, which avoids the insufficient efficiency or accuracy caused by too many or too few points in the traditional strategy where the expansion points are uniformly distributed. Compared to the SOAR, the ASOAR can deal with wide low/mid-frequency bands both for response and adjoint equations with high precision and efficiency. Numerical examples show the validation and effectiveness of the proposed method. Full article

In this paper, we analyze an optimal control problem with a mixed cost function, which combines a terminal cost at the final state and an integral term involving the state and control variables. The problem includes both state and control constraints, which adds complexity to the analysis. We establish a necessary optimality condition in the form of the maximum principle, where the adjoint equation is an integral equation involving the Riemann and Stieltjes integrals with respect to a Borel measure. Our approach is based on the Dubovitskii–Milyutin theory, which employs conic approximations to efficiently manage state constraints. To illustrate the applicability of our results, we consider two examples related to epidemiological models, specifically the SIR model. These examples demonstrate how the developed framework can inform optimal control strategies to mitigate disease spread. Furthermore, we explore the implications of our findings in broader contexts, emphasizing how mixed cost functions manifest in various applied settings. Incorporating state constraints requires advanced mathematical techniques, and our approach provides a structured way to address them. The integral nature of the adjoint equation highlights the role of measure-theoretic tools in optimal control. Through our examples, we demonstrate practical applications of the proposed methodology, reinforcing its usefulness in real-life situations. By extending the Dubovitskii–Milyutin framework, we contribute to a deeper understanding of constrained control problems and their solutions. Full article

This work presents the general mathematical frameworks of the “First and Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integral Equations of Volterra Type” designated as the 1st-FASAM-NIE-V and the 2nd-FASAM-NIE-V methodologies, respectively. Using a single large-scale (adjoint) computation, the 1st-FASAM-NIE-V enables the most efficient computation of the exact expressions of all first-order sensitivities of the decoder response to the feature functions and also with respect to the optimal values of the NIE-net’s parameters/weights after the respective NIE-Volterra-net was optimized to represent the underlying physical system. The computation of all second-order sensitivities with respect to the feature functions using the 2nd-FASAM-NIE-V requires as many large-scale computations as there are first-order sensitivities of the decoder response with respect to the feature functions. Subsequently, the second-order sensitivities of the decoder response with respect to the primary model parameters are obtained trivially by applying the “chain-rule of differentiation” to the second-order sensitivities with respect to the feature functions. The application of the 1st-FASAM-NIE-V and the 2nd-FASAM-NIE-V methodologies is illustrated by using a well-known model for neutron slowing down in a homogeneous hydrogenous medium, which yields tractable closed-form exact explicit expressions for all quantities of interest, including the various adjoint sensitivity functions and first- and second-order sensitivities of the decoder response with respect to all feature functions and also primary model parameters. Full article

Water residence time (WRT) is a crucial parameter for evaluating the rate of water exchange and it serves as a timescale for elucidating hydrodynamic processes, pollutant dispersion, and biogeochemical cycling in coastal waters. This study investigates the tidal-driven WRT patterns in the Bohai and Yellow Seas (collectively known as BYS) by employing a tidal model in conjunction with an adjoint WRT diagnostic model and explores the influence of tidal constituents on WRT. The findings indicate that the tidal-driven WRT in the BYS is approximately 2.11 years, exhibiting a significant spatially heterogeneous distribution. The WRT pattern shows a strong correlation with the pattern of tidal-driven Lagrangian residual currents (LRCs). Semidiurnal tides have a more pronounced effect on WRT than diurnal tides. Semidiurnal tides significantly reduce WRT across the entire BYS, while diurnal tides predominantly influence WRT in the Bohai Sea (BS). The M₂ tidal constituent is the most influential in decreasing WRT and enhancing water exchange, owing to its dominant energy contribution within the tidal system. In contrast, the S₂ tidal constituent has a minimal effect; however, its interaction with the M₂ tidal constituent plays a significant role in reducing the WRT. The K₁ and O₁ constituents exert more localized effects on WRT, particularly in the central BS, where their energy ratios relative to M₂ are relatively high. Although the amplitude of the S₂ constituent exceeds that of K₁ and O₁, its contribution to LRC—and consequently to WRT—is limited due to the overlapping tidal wave with M₂. This research contributes to a deeper understanding of the influence of tidal dynamics on long-term water transport and associated timescales, which are vital for enhancing predictions of material transport and ecosystem dynamics in tidal-dominated environments. Full article