The paper introduces a two‑stage physics‑informed deep learning framework for solving inverse problems in partial differential equations with jump discontinuities in coefficients. The first stage uses a dual‑network architecture to approximate the PDE solution and a relaxed continuous surrogate of the coefficient field, followed by Bayesian inference with Gaussian mixture and birth‑death Markov chain models to estimate coefficient regimes and transition regions. The second stage reformulates the inverse problem as a constrained estimator with a hard piecewise‑constant coefficient representation, achieving accurate parameter estimation with acceptable computational costs across various PDE types.
arXiv:2608. 04778v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) incorporate governing equations into neural-network training and can approximate PDE solutions without requiring large observational datasets.
arXiv:2508. 09156v3 Announce Type: replace-cross Abstract: We present a framework for fine-tuning flow-matching generative models to enforce physical constraints and solve inverse problems in scientific systems.
arXiv:2503. 05598v2 Announce Type: replace-cross Abstract: This review examines neural operator architectures for learning solution operators of parametric partial differential equations (PDEs), with an emphasis on conceptual clarity and practical implementation.
arXiv:2606. 12337v1 Announce Type: cross Abstract: Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization, while physics-informed neural networks (PINNs) have emerged as a flexible alternative.
The paper presents a method that combines flow‑matching models with energy‑based modeling to explicitly construct scalar energy functions for physical fields. These energies are derived from a matching regression objective on a linear Gaussian interpolation, avoiding variational approximations or extra MCMC steps, and can be used for energy‑corrected data generation, out‑of‑distribution detection, and posterior sampling in inverse problems. The approach enables general MCMC samplers that reduce PDE residuals and spectral distance, and it demonstrates that combining data‑driven and physics‑based energies improves OOD detection accuracy.
The paper introduces a two‑step debiased estimation method for PDE‑constrained inverse problems where the PDE solution is approximated by Physics‑Informed Neural Networks (PINNs). By combining neural‑network‑based nonparametric estimation with an influence‑function bias correction, the authors achieve a √{n}-consistent, asymptotically normal estimator without undersmoothing the neural network. The approach is extended to Bayesian inference, yielding a posterior that contracts at the √{n}-rate with asymptotic covariance matching the frequentist estimator, and the analysis also provides near‑minimax rates for estimating nonparametric regression functions and their derivatives in Sobolev spaces.
arXiv:2607. 20378v1 Announce Type: new Abstract: Physics-informed learning of partial differential equations (PDEs) has been dominated by multilayer perceptrons (MLPs), whose spectral bias and dense parameterization limit both accuracy and interpretability.
arXiv:2505. 22391v2 Announce Type: replace-cross Abstract: Modeling physical systems in a generative manner offers several advantages, including the ability to handle partial observations, generate diverse solutions, and address both forward and inverse problems.
arXiv:2602. 09708v2 Announce Type: replace-cross Abstract: We propose physics-informed spectral diffusion (PISD), a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of partial differential equations (PDEs) conditioned on partial observations, which includes, in particular, forward and inverse PDE problems.
The paper presents a method to recover unknown functional terms in partial differential equations (PDEs) by embedding neural networks into standard parameter estimation workflows. By training on data, the approach learns interaction kernels and external potentials in nonlocal aggregation‑diffusion equations, achieving high accuracy. The study systematically investigates how reconstruction accuracy depends on solution diversity, sampling density, and measurement noise.
The paper evaluates the robustness of Physics‑Informed Neural Networks (PINNs) against noisy data in inverse problems, comparing them to a finite element method (FEM) plus optimizer baseline. Experiments on viscosity identification in 1D Burgers’ equation and 2D/3D Taylor‑Green Vortex with additive Gaussian noise show that PINNs, while requiring less human expertise, are outperformed by the traditional FEM approach in accuracy (e.g., RMSE 0.01 vs. 0.0013 for 2D Taylor‑Green with σ=1). PINNs do, however, exhibit better scaling with problem complexity, and the study highlights specific training failures that must be addressed for PINNs to become more competitive.
arXiv:2608. 11435v1 Announce Type: new Abstract: Forward and inverse modeling of parametric dynamical systems requires surrogate models that are not only accurate for state prediction, but also informative for parameter calibration.