physics-informed loss

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.

This work addresses the poor robustness and low accuracy of physics-informed neural networks (PINNs) in solving oscillatory, multiscale, and long-time-evolving partial differential equations (PDEs). We propose the Physics- and Equation-Constrained Artificial Neural Network (PECANN), an enhanced framework for PDE learning. Key contributions include: (1) a condition-adaptive penalty update strategy enabling cooperative optimization of multiscale physical constraints; (2) pointwise expectation-form constraints coupled with a sliding temporal window mechanism to improve training stability and long-term predictive capability; and (3) an efficient end-to-end solver integrating the augmented Lagrangian method, Fourier feature mapping, mini-batch training, and terminal-state propagation. Evaluated on challenging benchmarks—including transonic flow, high-wavenumber Helmholtz equations, and spatially varying heat source inversion—PECANN achieves accuracy comparable to state-of-the-art numerical solvers and significantly outperforms existing PINN variants, demonstrating superior robustness, generalizability, and practical applicability.

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.

Whether the cost function must be embedded into neural network training for optimal control remains an open question. Method: We propose a decoupled paradigm: first, independently train neural operators (e.g., DeepONet) using PDE residual penalties to model the underlying physical system; then, perform online optimization of control variables via automatic differentiation and unconstrained optimizers (e.g., L-BFGS), fully excluding the cost function from the training phase. Contribution/Results: We provide the first theoretical and empirical validation that the cost function can be entirely removed from training—enabling “one-time physics modeling + multi-objective online optimization.” Using only three DeepONet models, we achieve high-accuracy solutions across nine distinct optimal control problems. The framework exhibits strong generalization and consistency under varying cost functions, significantly simplifying architecture design and enhancing deployment flexibility.

To address training failure in Physics-Informed Neural Networks (PINNs) caused by gradient direction conflicts among multi-task loss components, this work theoretically establishes—under a second-order optimization perspective—that Hessian preconditioning inherently alleviates such conflicts. Building on this insight, we propose SOAP, a quasi-Newton method that efficiently approximates the Hessian preconditioner, and introduce a Multi-Gradient Alignment Score (MGAS), an extended cosine similarity metric quantifying alignment across task gradients. Evaluated on ten challenging PDE benchmarks, SOAP achieves state-of-the-art performance: it is the first method to stably solve turbulent flow problems at Reynolds numbers up to Re = 10,000, and delivers 2–10× higher accuracy than existing approaches. These advances substantially broaden the applicability of PINNs to strongly nonlinear, complex physical systems.

This work addresses the persistent challenge in physics-informed neural networks (PINNs) where conflicting gradients between PDE residuals and boundary constraints often trap optimization in poor local minima. The study systematically uncovers, for the first time, the underlying causes of this gradient pathology and introduces the Constraint Alignment and Manifold Lifting (CAML) framework. CAML mitigates gradient conflicts by reformulating zeroth-order terms into geometrically agnostic alignment constraints and incorporates a delay factor to steer optimizers away from high-curvature regions of the loss landscape. By transcending the limitations of existing adaptive weighting and hard-constraint strategies, the proposed method substantially enhances numerical stability and solution efficiency across complex PINN problems. The implementation is publicly released to facilitate reproducibility and further research.

This work addresses the low sample efficiency and inconsistent actions often observed in reinforcement learning for robotic control, which stem from neglecting known physical dynamics. To this end, the authors propose PIPER, a novel framework that seamlessly integrates physical priors into policy learning by incorporating a differentiable Lagrangian dynamics residual—computed via a standard simulator—as a soft regularization term directly into the policy objective. Crucially, this approach requires no modifications to the underlying simulator or reinforcement learning algorithm. By softly enforcing analytical physical constraints during policy updates, PIPER achieves a tight coupling between physical consistency and learning, significantly improving sample efficiency, training stability, and control accuracy. Empirical results across multiple robotic tasks demonstrate that policies trained with PIPER exhibit superior physical plausibility and overall performance compared to baseline methods.

This work addresses the challenge of ensuring safety in industrial cyber-physical systems when applying deep reinforcement learning, where black-box exploration may inadvertently violate hardware constraints and conventional reward shaping struggles to balance safety with task performance. To overcome this, the authors propose a physics-informed safety mechanism that embeds a differentiable dynamics model into the loss function of a Proximal Policy Optimization (PPO) policy network. By performing short-horizon forward simulations to predict trajectories, the method imposes soft penalties—decoupled from the task-specific reward—on potential safety violations. This approach regularizes the policy online without requiring intricate reward engineering. Evaluated on a one-degree-of-freedom helicopter simulation platform, the method significantly reduces pitch angle constraint violations while maintaining excellent trajectory tracking performance.

This work addresses key limitations of conventional physics-informed neural networks (PINNs) in solving differential equations—namely parameter redundancy, weak locality, and insufficient control over solution smoothness. The authors propose Physics-Informed Splines (PI-Splines), which directly parameterize the unknown field using tensor-product B-splines with trainable control points. Retaining the residual-driven training paradigm of PINNs, PI-Splines inherently offer compact support, explicit smoothness control, and analytically computable derivatives. By strongly enforcing boundary conditions, the method endows spline parameters with clear geometric meaning while leveraging structured representation and efficient optimization strategies. Experimental results demonstrate that PI-Splines achieve stable and efficient performance across multiple benchmark problems, significantly outperforming traditional neural network architectures in terms of parameter efficiency, locality, and representational capacity.

This work addresses the challenge of efficiently adapting pre-trained foundation models for partial differential equations (PDEs) under data scarcity and distribution shift. We propose a physics-informed fine-tuning framework that incorporates physical constraints—such as PDE residuals and boundary conditions—directly into the fine-tuning objective, enabling data-efficient adaptation without requiring ground-truth solutions. To our knowledge, this is the first systematic demonstration of the effectiveness of physics-informed fine-tuning for transferring PDE foundation models. By integrating a hybrid fine-tuning strategy, our approach significantly enhances out-of-distribution generalization. Experiments show that, even in the absence of ground-truth solutions, our method achieves accuracy comparable to purely data-driven approaches on unseen PDE tasks and consistently outperforms them when only limited data are available.