physics-informed learning

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 lack of a unified understanding of the design principles, applicability, and performance differences between Physics-Informed Neural Networks (PINNs) and Neural Operators (NOs), which hinders the development of reliable data-driven PDE solvers. It proposes the first unified analytical framework that systematically characterizes the design space of both approaches along three dimensions: learning objectives, mechanisms for embedding physical structure, and strategies for computational load distribution. By elucidating the intrinsic connections and fundamental distinctions between these methods, the study not only clarifies the positioning and performance origins of existing techniques but also provides theoretical guidance and novel pathways for designing efficient and robust PDE solvers that effectively integrate physical priors with data-driven learning.

Modeling dynamical systems faces challenges including data scarcity, high uncertainty, poor interpretability, and unreliable predictions. Method: This paper proposes a novel Physics-Enhanced Machine Learning (PEML) paradigm that unifies conceptual foundations, systematically categorizes approaches into physics-guided, physics-encoded, and physics-constrained methods, and identifies their applicability boundaries and reliability mechanisms. It introduces four types of physics- and domain-knowledge-induced biases to characterize modeling error sources. Methodologically, PEML integrates partial differential equation constraints, conservation law embedding, uncertainty propagation modeling, and interpretability-driven neural architectures. Contribution/Results: Experiments demonstrate that PEML significantly improves accuracy, robustness, and trustworthiness of long-term forecasting and inverse inference under small-data regimes. The framework provides a scientifically grounded yet practically deployable modeling tool for high-consequence engineering decision-making.

This study addresses critical challenges in applying deep learning to scientific computing—namely, poor interpretability, heavy data dependency, and insufficient physical consistency—within physics-based simulation scenarios. We propose a physics-driven AI modeling framework integrating physics-informed loss functions, differentiable simulators, diffusion-based generative models, physics-guided reinforcement learning, and custom neural architectures, implemented via an interactive Jupyter-based experimental platform. Crucially, we pioneer the systematic embedding of physical priors across the entire deep learning pipeline—model formulation, training, and inference—enabling high-fidelity, data-efficient, and verifiable scientific modeling. The resulting methodology is modular, reusable, and immediately deployable, significantly enhancing model generalizability and interpretability. This work establishes a novel paradigm and technical foundation for next-generation scientific foundation models.

Sensitivity analysis for steady-state heat conduction in heterogeneous materials—characterized by strong phase contrast and temperature-dependent properties—is computationally expensive when performed via conventional adjoint methods. Method: This paper proposes the Finite Operator Learning (FOL) framework, which tightly integrates neural operators with finite element discretization. FOL embeds physical constraints—including the weak-form energy functional, boundary conditions, and residual stationarity—into a multi-objective loss function, and combines Sobolev-norm training with feedforward networks to jointly predict both PDE solutions and their sensitivities to design parameters in an end-to-end manner. Contribution/Results: FOL requires neither labeled training data nor adjoint computations, ensuring strong physics consistency. It directly outputs high-fidelity solutions and accurate gradients, enabling tangent-matrix-driven microstructural thermal optimization. By eliminating iterative adjoint solves, FOL significantly reduces sensitivity analysis cost while preserving numerical robustness and physical fidelity.

This work addresses the significant limitations in accuracy and convergence of Physics-Informed Neural Networks (PINNs) when training data are inconsistent with the governing partial differential equation (PDE). The authors introduce the concept of a “consistency barrier,” which reveals an inherent lower bound on error arising from the tension between data fidelity and PDE residual enforcement. Through controlled experiments on the 1D viscous Burgers equation—leveraging analytical solutions, multi-fidelity numerical data, and residual-driven training—they demonstrate that PINN errors under low-fidelity data are fundamentally constrained by this consistency barrier. In contrast, when high-fidelity data are employed, the barrier is effectively eliminated, yielding PINN solutions statistically indistinguishable from the analytical solution. This study provides both theoretical insight and empirical evidence to advance the understanding and robustness of PINNs in practical applications.

This work addresses the challenge of constructing efficient surrogate models for parametrized systems in multi-query scenarios—such as optimization, control, and uncertainty quantification—by proposing a unified scientific machine learning framework that systematically integrates physics-driven, data-driven, and hybrid modeling paradigms. The framework encompasses techniques including Proper Orthogonal Decomposition (POD), Proper Generalized Decomposition (PGD), and neural networks, viewed through the lens of function approximation. A key innovation lies in unifying the selection of reduced-order bases and approximation criteria within a coherent analytical framework. Furthermore, the study explores emerging directions such as multi-fidelity fusion, adaptive sampling, and data augmentation. The resulting methodology offers both theoretical foundations and novel modeling paradigms with broad applicability in digital twins, smart manufacturing, and personalized medicine.

Traditional physics-informed neural networks (PINNs) for solving partial differential equations often neglect the irreversibility mandated by the second law of thermodynamics, leading to unphysical solutions or training failure. To address this, we propose a general and robust irreversibility regularization strategy: soft constraints—based either on entropy production rate or temporal monotonicity—are incorporated into the PINN loss function without altering the network architecture, requiring only minimal adaptation. This work is the first to explicitly embed thermodynamic irreversibility into the PINN training process, thereby enforcing the unidirectional evolution characteristic of physical processes. Evaluated across diverse irreversible phenomena—including traveling wave propagation, combustion, ice melting, corrosion, and crack propagation—the method reduces prediction errors by over one order of magnitude. It significantly enhances both physical consistency and training stability of learned solutions.

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.

In physics-informed machine learning, conventional Sobolev minimax rates are constrained by data dependence structures, limiting achievable learning rates under non-i.i.d. settings. Method: We propose a physics-regularized empirical risk minimization framework that explicitly incorporates accurate physical priors into the loss function, enabling improved learning rates under dependent data. Contribution/Results: We theoretically establish that when the embedded physical model exactly matches the underlying governing law, the excess risk converges at the optimal i.i.d. rate—accelerating beyond standard Sobolev rates—without compromising sample efficiency. This is the first result to quantify the convergence-rate acceleration induced by physical priors in non-i.i.d. regimes. Our analysis provides a novel generalization-theoretic paradigm for physics-informed learning, bridging physical modeling fidelity with statistical learning guarantees under data dependence.