The paper investigates when inexpensive low‑fidelity (LF) physics models can reduce the number of costly high‑fidelity (HF) simulations needed for accurate solutions of parametric partial differential equations. It derives a sample‑complexity bound showing that learning the HF field via its discrepancy from the LF model can cut HF samples by a factor of κ^{2γ}, where κ is the complexity ratio and γ the effective‑dimension exponent, and confirms this scaling in a kernel setting. Guided by this theory, the authors introduce GaMF‑PINN, a gated multi‑fidelity physics‑informed neural network that selectively trusts the LF model and falls back to physics‑only learning when necessary, coupled with a conformal uncertainty procedure that guarantees coverage of the HF solution. The method is evaluated on the newly released MFPDE‑Bench, achieving HF‑PINN accuracy with up to eight times fewer HF solves, robustness to corrupted LF models, and reliable uncertainty estimates compared to Bayesian PINNs and deep ensembles.
arXiv:2606. 00643v1 Announce Type: cross Abstract: Physics-Informed Neural Networks (PINNs) often train slowly or fail to converge on challenging partial differential equations (PDEs), a behavior recently linked to severely ill-conditioned loss landscapes inherited from the underlying differential operator.
arXiv:2607. 02194v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) have emerged as a promising route to solve partial differential equations, yet they have struggled to reach the precision of classical solvers.
arXiv:2606. 09923v1 Announce Type: cross Abstract: Neural operators such as the Fourier Neural Operator (FNO) have emerged as powerful surrogates for solving partial differential equations (PDEs), achieving speedups of several orders of magnitude over traditional numerical solvers.
Physics-Informed Conformal Prediction (PI‑CP) embeds PDE residuals into the nonconformity score of split conformal prediction, yielding distribution‑free prediction intervals with provable coverage that adapt spatially to physics violations. The method demonstrates consistent 89‑91% coverage across six physics scenarios, outperforming MC Dropout and Deep Ensembles, while Fourier Neural Operators (FNO) achieve superior accuracy over CNN and DeepONet. Additionally, the authors prove that FNO’s translation equivariance limits its ability to solve PDEs with Dirichlet boundary conditions, and show that adding coordinate channels can reduce error by up to 63×.
arXiv:2609. 16406v1 Announce Type: cross Abstract: Machine learning-based partial differential equations (PDEs) solvers have attracted significant attention in recent years.
arXiv:2609.24947v1 Announce Type: new Abstract: Neural operators evaluate parametric partial differential equations cheaply but degrade sharply outside their training distribution. Physics-informed n...
arXiv:2605. 08318v2 Announce Type: replace Abstract: We study the problem of \emph{architecture selection} for deep learning models trained to solve partial differential equations (PDEs), asking when transformer-based architectures with learned attention outperform Fourier-domain neural operators.
arXiv:2607. 15702v1 Announce Type: cross Abstract: We prove a finite-sample formulation gap for physics-informed learning of nonlinear multiscale elliptic equations.
arXiv:2606. 24999v1 Announce Type: new Abstract: High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging.
arXiv:2605. 23391v2 Announce Type: replace Abstract: Physics-informed neural networks (PINNs) for coupled multiphysics systems suffer systematic accuracy degradation as inter-equation coupling strengthens.
arXiv:2607. 15702v2 Announce Type: replace-cross Abstract: We develop a non-asymptotic approximation, sampling, and finite-iteration optimization theory for variational physics-informed approximation of uniformly monotone nonlinear multiscale elliptic equations.
The paper introduces S$^{2}$-PINN, a stochastic separable physics-informed neural network that models the solution of random partial differential equations using a learnable Gaussian spatial dictionary, Fourier temporal features, and a generalized polynomial chaos stochastic basis linked by a low-rank Canonical Polyadic tensor core. Training employs a hybrid strong-form and gPC-projected residual loss, and theoretical analysis shows the separable class is dense in $L^2$ and that the projected residual matches a stochastic Galerkin constraint. Experiments on four manufactured benchmarks and additional Poisson, Darcy, Navier–Stokes, and inverse problems demonstrate that S$^{2}$-PINN achieves higher mean and variance accuracy, better calibration, and fewer parameters than nine baselines, supporting stochastic separability as an effective design principle for physics-informed neural UQ.