enforce energy conservation

This paper addresses three physics-constrained regression problems in fluid mechanics: PIV velocity field super-resolution and data assimilation, data-driven turbulence modeling, and system identification for digital twin predictive control. Methodologically, it proposes a physics-informed regression framework that incorporates conservation laws—such as the Navier–Stokes equations—as soft constraints into supervised learning objectives; gradient-based optimization is enabled via automatic differentiation, and differentiable physics-informed models are implemented in Python. Key contributions include: (1) a unified approach to modeling under multiscale dynamics, limited data, and high noise; (2) substantially improved model generalizability and physical consistency; and (3) publicly available, reproducible educational case studies and code, demonstrating the efficacy and pedagogical versatility of physics-informed learning in scientific discovery and engineering closed-loop control.

Physics-informed neural networks (PINNs) enforce partial differential equations (PDEs) via soft constraints, failing to guarantee conservation of linear and quadratic integral quantities—compromising physical consistency and numerical accuracy. This work proposes a novel projection-based method that strictly enforces either independent or joint conservation of these integrals during training. By formulating and solving a nonlinear constrained optimization problem, we derive an explicit projection operator that orthogonally projects the neural network output onto the corresponding conservation manifold in real time. To our knowledge, this is the first approach enabling configurable, simultaneous control of both linear and quadratic integral conservation. The method significantly improves the condition number of the loss landscape, enhancing training stability and convergence speed. Experiments demonstrate reductions in conservation error by three to four orders of magnitude, accompanied by commensurate decreases in PDE solution error, markedly improving physical fidelity and generalization capability.

Neural networks solving partial differential equations (PDEs) often violate fundamental physical principles—such as mass conservation, entropy production, positivity, and shock dynamics. To address this, we propose the Constraint Projection Learning (CPL) framework, which encodes conservation laws, the Rankine–Hugoniot condition, entropy conditions, and positivity constraints into differentiable projection operators embedded throughout training. Integrated with total variation diminishing (TVD) regularization and rollout curriculum learning, CPL enforces strict physical compliance at every optimization step. Experiments on the Burgers and Euler equations demonstrate that solutions exhibit exact conservation, bounded total variation, no error accumulation, long-term stability, and machine-precision adherence to physical laws. This work presents the first end-to-end, differentiable, and compact integration of multiple physics-based constraints in neural PDE solvers, significantly enhancing solution reliability and generalizability.

This study addresses the challenge that PDE state constraints in engineering design induce highly anisotropic feasible regions, rendering traditional Bayesian optimization inefficient. To overcome this, we propose linearly mapping state-space constraints into the design space and constructing ellipsoids to generate diverse candidate points while updating surrogate models online. Furthermore, we introduce precomputed control sets and a dynamic ellipsoid reconstruction strategy to effectively mitigate anisotropic sampling bottlenecks. The proposed method is validated end-to-end on tokamak divertor design, where it significantly enhances performance while maintaining plasma boundary states. This work establishes an efficient Bayesian optimization paradigm for engineering design problems governed by complex PDE constraints.

This work addresses the lack of systematic methodologies in model optimization, which often relies on heuristic choices and struggles to accommodate diverse deployment constraints. It formalizes model compression and acceleration as a constraint-aware multi-objective engineering decision problem, establishing a unified and actionable framework grounded in five key dimensions: data availability, latency, memory footprint, accuracy tolerance, and retraining budget. By integrating techniques such as quantization, pruning, knowledge distillation, parameter-efficient fine-tuning (PEFT), and inference optimization, the study proposes tailored optimization pipelines for four representative industrial scenarios, delivering a reproducible and quantifiable guide for technology selection.

This work addresses the practical limitations of expensive yet powerful propagators—such as Energetic Reasoning—in constraint programming, which, despite their strong pruning capabilities, incur substantial computational overhead. To mitigate this issue, the paper proposes a hybrid framework that integrates static machine learning with dynamic search heuristics to control propagator activation. Specifically, a supervised learning approach is employed to construct a predictor function that dynamically decides whether to invoke the costly propagator during search. The authors design a set of effective instance-specific features and train a high-accuracy classification model, achieving, for the first time, seamless integration of such a predictive mechanism into a modern constraint solver. Experimental results demonstrate the feasibility of the approach and shed light on key challenges and design principles for building efficient propagator selection strategies.

This study addresses the high computational overhead, excessive memory consumption, and limited robustness of existing constraint satisfaction methods by proposing the Projection-Adaptive Loss (PAL) framework. Departing from conventional multi-step unfolding paradigms, PAL integrates deep learning with a projected gradient method, enforcing nonlinear constraints during training through a single decoupled projection step coupled with an adaptive weighting mechanism. Experimental results demonstrate that PAL accelerates training by 2.5× while maintaining near-perfect constraint feasibility under extreme nonlinearity. By significantly outperforming established baselines, this work achieves an effective unification of resource efficiency and solution accuracy.

研究解决了物理模型在条件变化时的两种失效问题，通过使用辛积分器保持长期稳定性，通过显式线性因子化实现反事实迁移。

研究通过比较MSE和AL训练方法，探讨了如何在扩大电网规模时减少ACOPF代理学习中的物理约束违反问题。

This study investigates the training dynamics of coupled learning (CL) and equilibrium propagation (EP) in the continuous-time, small-perturbation limit, revealing a parameter conservation law in physically realizable systems that parallels mass conservation. By leveraging continuous-time dynamical systems analysis and perturbation theory, the work establishes— for the first time—that this conservation law holds universally across a broad range of physical settings. Furthermore, it elucidates how this constraint governs the convergence behavior of learning in linear circuits. The findings not only enhance the reliability of CL and EP training but also provide a rigorous theoretical foundation and practical guidance for achieving efficient and stable learning in neuromorphic hardware implementations.