energy conservation constraints

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.

Under energy transition, high renewable penetration and cross-border interconnections intensify grid uncertainty, rendering conventional power flow solvers inadequate for real-time operation, while purely data-driven models lack physical consistency. This paper proposes a physics-informed neural network (PINN) framework for power flow simulation, integrating Kirchhoff’s laws and other physical priors via hybrid modeling—combining MLP/GNN architectures with physics-constrained regularization and unsupervised loss. We introduce a novel four-dimensional evaluation framework (accuracy, physical consistency, industrial deployability, out-of-distribution generalization) and the LIPS benchmark platform. Systematic ablation studies demonstrate that explicit graph-structured modeling and direct optimization of physical equations are critical to reliability. The resulting model achieves high accuracy, strong physical consistency, robust out-of-distribution generalization, and practical industrial deployment potential. Code is fully open-sourced.

Addressing the dual challenges of physical inconsistency and poor generalization under limited data, this work introduces a novel paradigm that projects model outputs onto a physics-defined manifold. Specifically, predictions are explicitly projected onto a differential-geometric manifold constrained by prior physical laws—such as conservation principles—ensuring inherent compliance with physical constraints. The method is architecture-agnostic and task-agnostic, overcoming the unreliable generalization of penalty-based approaches and the inflexibility of physics-invariant architectures. By integrating implicit constraint enforcement (e.g., via Lagrange multipliers) with a plug-and-play interface, it seamlessly interoperates with Physics-Informed Neural Networks (PINNs). Evaluated on benchmark tasks—including a spring-mass oscillator and low-temperature reactive plasma modeling—the approach reduces physical law violation rates by 92% and decreases prediction error of key state variables by 37%. Under scarce-data regimes, it significantly outperforms both standard PINNs and purely data-driven models.

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 investigates whether physical conservation laws can be rigorously certified in learned latent representations and quantifies the temporal horizon over which such conservation remains valid in roll-out predictions. To this end, the authors introduce the notion of a "certified time horizon" and develop a certification framework centered on decoding physical invariants. Prediction error is decomposed into contributions from representation, readout, and latent dynamics. A monotonic alignment between soft and hard certification is established via shell-like temporal certificates and soft witness functions. The approach integrates Lipschitz control, geometric priors (e.g., symplectic structure), and invariant decoding, and is validated across state-space, learned coordinates, and pixel-level observations. Experiments show that hard symplectic constraints excel in known phase spaces but lack generalizability to learned coordinates, whereas soft alignment proves more robust in learned representations; the Kepler problem delineates the limits of geometric constraints, and pixel-level certification is recoverable within stable readout sub-tubes.

Neural networks often struggle to strictly satisfy nonlinear constraints during inference, which hinders their deployment in safety-critical applications. This work proposes HardNet++, the first method capable of enforcing hard satisfaction of general nonlinear equality and inequality constraints, overcoming the limitation of existing approaches that are restricted to specific constraint forms. By integrating damped local linearization, differentiable projection layers, and end-to-end training, HardNet++ guarantees constraint compliance simultaneously during both training and inference. Evaluated on model predictive control tasks, HardNet++ achieves high-precision constraint adherence while preserving solution optimality.

This work investigates whether intrinsic symmetries in training data induce conserved quantities during gradient flow training of neural networks. By integrating tools from differential geometry and dynamical systems theory, the study establishes—for the first time—a systematic connection between data symmetries and conservation laws in training dynamics, employing tensorized networks (including linear, polynomial, and Lightning Attention architectures) as an analytical framework. The analysis demonstrates that, under general non-polynomial losses, data symmetries do not yield additional conserved quantities; however, when combined with data augmentation under mean squared error (MSE) loss, novel conserved quantities emerge. This finding uncovers a distinctive conservation mechanism specific to MSE loss and offers a new perspective for understanding the dynamics of neural network training.

This work proposes a novel Bayesian framework that explicitly incorporates linear equality constraints—encoding known physical laws—into variational Bayesian inference, thereby jointly optimizing physical consistency and uncertainty quantification. By integrating Bayesian neural networks with a constraint-embedding mechanism, the method enables unified uncertainty modeling over both model parameters and domain knowledge. Evaluated on a single-particle battery modeling task, the approach significantly narrows predictive credible intervals and drastically reduces violations of the prescribed linear constraints compared to standard variational Bayesian neural networks, demonstrating its effectiveness and practical utility in delivering reliable, physics-informed predictions with well-calibrated uncertainty estimates.

This work addresses the role of physical constraints in autoregressive scientific forecasting, where constraints are commonly enforced by correcting predicted states—a practice whose efficacy hinges on the accuracy of the constraint operator. The study proposes “operator–data alignment” as a principled criterion to systematically evaluate constraint strategies. Experiments demonstrate that exact projections, such as Fourier-based projection, substantially improve rollout accuracy (reducing MSE from 9.39×10⁻⁵ to 5.37×10⁻⁷ on NS-128), whereas approximate corrections like Poisson cleaning induce distributional shifts in non-periodic flows and degrade performance. Through hierarchical prediction, adaptive gating, and external backbone validation on Navier–Stokes and CFDBench benchmarks, the authors show that target distortion—measured by MSE relative to the ground truth—is a more reliable indicator of performance degradation than residual error, underscoring the critical importance of geometric alignment for effective constraint enforcement.

This work addresses the challenge of enforcing physical symmetries—such as rotational equivariance—in machine learning models without imposing explicit architectural constraints. The authors propose a general, architecture-agnostic approach that introduces a novel metric to quantify the degree of symmetry learning, employs spectral analysis to diagnose failure modes, and leverages targeted data augmentation to guide unconstrained Transformers—including graph neural networks and PointNet-style architectures—to progressively approximate equivariance across layers during training. Experiments demonstrate that injecting only the minimal necessary inductive bias substantially enhances physical fidelity, numerical stability, and predictive accuracy, while preserving the model’s expressive capacity.