physics-informed modeling

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.

Accurate forecasting of multivariate time series governed by unknown partial differential equations (PDEs) remains challenging under sparse, noisy, or incomplete observational data. Method: We propose an end-to-end, data-driven framework that jointly discovers governing PDEs and builds physics-constrained models. Leveraging symbolic regression integrated with sparse identification of nonlinear dynamics (SINDy), we automatically infer PDE structure and embed it uniformly into three distinct modeling paradigms: physics-informed neural networks (PINNs), Bayesian PINNs, and Bayesian linear regression. Joint optimization of physics-informed losses and Bayesian objectives enables simultaneous inference of PDE parameters and predictive uncertainty. Contribution/Results: This work achieves the first seamless integration of automated PDE discovery with multi-paradigm physics-guided learning—requiring no prior knowledge of the underlying PDE. Experiments on real-world multivariate time-series datasets demonstrate substantial improvements in both forecast accuracy and uncertainty calibration, particularly under low-quality data conditions, outperforming purely data-driven baselines.

Existing physics-informed machine learning methods for digital twin modeling suffer from either shallow physical integration or poor architectural generalizability. To address this, we propose an encoding-based residual network that tightly couples physics operators with learnable modules. Our approach introduces plug-and-play physics encoding blocks—grounded in analytical models such as the Euler–Lagrange equations—and integrates them with intermediate residual connections to explicitly embed geometric and kinematic constraints. Leveraging hybrid loss optimization and joint data-physics training, the framework achieves high physical fidelity and strong generalization under low-data and low-parameter regimes. Evaluated on robotic motion modeling and autonomous vehicle steering simulation, our method reduces parameter count by over 40% and decreases small-sample prediction error by more than 35% compared to both pure data-driven baselines and state-of-the-art physics-informed neural networks (PINNs).

Physics-informed neural networks (PINNs) face challenges—including ill-conditioned optimization, slow convergence, and poor generalization—when solving parametric partial differential equations (PDEs). This paper proposes a data-driven neural solver that parameterizes adaptive gradient descent as a neural network, jointly modeling distributions of PDE coefficients and initial/boundary conditions under physical constraints, while dynamically conditioning the optimizer to alleviate loss function ill-conditioning. To our knowledge, this is the first work to introduce neural solvers into parametric PDE settings, enabling end-to-end training via implicit differentiation and backpropagation. Experiments demonstrate a 2–5× speedup in training with enhanced convergence stability. At inference, the solver generalizes robustly to unseen parameter combinations, significantly reducing required iterations while maintaining high accuracy.

This study addresses the challenge of jointly discovering physically meaningful concepts and their governing equations—where physical principles and differential equations are tightly coupled and cannot be identified independently. We propose a unified end-to-end discovery framework that integrates variational autoencoders (VAEs) with neural ordinary differential equations (Neural ODEs), embedding differentiable physics-informed priors and emulating human-like physical reasoning inspired by the SciNet architecture. The framework simultaneously learns interpretable latent physical concepts and their exact mathematical formulations from simulation data. Crucially, it enables joint differentiable optimization of both concept representations and equation structures—overcoming limitations of conventional symbolic regression and black-box fitting. Evaluated on canonical problems—including heliocentrism, Newtonian gravitation, the Schrödinger equation, and the Pauli magnetic moment—the model recovers theoretically correct functional forms; learned concepts carry clear physical interpretations; and predictive errors remain below 1%.

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 limited data efficiency and poor out-of-distribution (OOD) generalization of existing neural operator methods, which often neglect underlying physical principles—particularly when facing parameter variations or simulation-to-reality transfer. To overcome these limitations, we propose a multi-physics joint training framework that explicitly integrates the original partial differential equations (PDEs) with their simplified canonical forms directly into the neural operator training process. This architecture-agnostic approach is compatible with diverse neural operator designs and consistently enhances model robustness under parameter shifts and cross-domain scenarios. Extensive experiments across multiple 1D, 2D, and 3D PDE tasks demonstrate significant reductions in normalized root mean square error (nRMSE), confirming improved data efficiency and superior OOD generalization performance.

Traditional PDE solvers are computationally expensive, while existing learning-based solvers struggle with optimization in stiff, multiscale, or large-domain problems and fail to adequately capture uncertainty propagation. This work proposes “flow learners,” which directly model the continuous evolution between physically admissible states by parameterizing transport vector fields and integrating them to generate trajectories. Grounded in optimal transport theory, the approach seamlessly integrates physical constraints with generative modeling, yielding significant improvements over current learning-based solvers in continuous-time prediction, native uncertainty quantification, and long-term simulation of complex dynamics. The method establishes a new pathway toward building physics-aware PDE solution frameworks.

This work addresses the longstanding challenge in nonlinear system identification of balancing physical interpretability with model flexibility: conventional approaches are constrained by fixed parametric forms, while neural ordinary differential equations (Neural ODEs) often lack physical grounding. To overcome this, we propose a state-dependent second-order quasi-linear parameter-varying (quasi-LPV) neural surrogate model that leverages state-conditioned modeling and a local physics-informed prompting mechanism. This framework reformulates system identification as a parameter manifold learning problem without requiring a predefined global dynamical equation, effectively decoupling trajectory reconstruction from physical parameter estimation to prevent optimization collapse. By integrating recurrent curriculum learning with a windowed ridge regression anchoring strategy, our method accurately recovers key physical parameters—such as natural frequencies, damping ratios, and gains—from sparse data and generates dynamics-consistent predictions, outperforming existing inverse modeling techniques across multiple benchmarks.

This work proposes a unified framework that integrates hierarchical Bayesian inference with data-driven closure learning to address the inverse problem of model calibration in multiphysics systems, where unknown parameters and incomplete dynamical laws pose significant challenges. The approach jointly infers system-specific parameters and shared unknown dynamics across multiple related systems through a hierarchical structure. Neural networks—such as Fourier Neural Operators (FNOs) and parameterized Physics-Informed Neural Networks (PINNs)—are employed to construct closure models for ODEs/PDEs. Efficient posterior inference is achieved via maximum marginal likelihood estimation combined with ensemble Metropolis-adjusted Langevin algorithm (MALA) sampling. Furthermore, an adaptive surrogate forward model and a bilevel optimization strategy are introduced to substantially reduce the computational cost associated with repeated forward solves. Experiments demonstrate that the framework achieves high calibration accuracy while enabling efficient joint modeling and computational acceleration across systems.