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.
arXiv:2603. 29237v2 Announce Type: replace Abstract: Enforcing prescribed global integral constraints in mesh-free neural PDE solvers is challenging in high-dimensional domains.
arXiv:2606. 18175v1 Announce Type: cross Abstract: We present a numerical method for the forward solution of nonlinear partial differential equations (PDEs) in which Bellman-Kalaba quasilinearization reduces the nonlinear problem to a sequence of linear subproblems, each discretized by collocation onto a trial space that is linear in its parameters and solved by a single direct linear least-squares QR factorization.
arXiv:2607. 06348v1 Announce Type: new Abstract: We introduce a physics-informed framework for learning finite-dimensional embeddings of solution families of partial differential equations.
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:2609.36165v1 Announce Type: cross Abstract: In this work, we develop a second-order optimization framework for physics-informed neural networks (PINNs) applied to high-dimensional parametric pa...
arXiv:2607. 13566v1 Announce Type: cross Abstract: For low-dimensional problems ($d\leq3$), spectral methods can achieve exceptionally high accuracy.
arXiv:2609. 16406v1 Announce Type: cross Abstract: Machine learning-based partial differential equations (PDEs) solvers have attracted significant attention in recent years.
We introduce a physics-informed framework for learning finite-dimensional embeddings of solution families of partial differential equations. The method uses a multihead Physics-Informed Neural Network in which a shared body learns a latent manifold representing the solution space, while linear heads reconstruct individual solutions associated with different initial conditions.
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:2608.24049v1 Announce Type: new Abstract: Neural operators provide efficient surrogates for spatiotemporal PDE systems, but purely data-driven formulations often accumulate substantial errors d...
arXiv:2605. 03542v2 Announce Type: replace-cross Abstract: The dual norm characterisation of weak solutions of second-order linear elliptic partial differential equations is mathematically natural but computationally intractable: evaluating the $H^{-1}$ norm of the residual requires a supremum over an infinite-dimensional test space.
arXiv:2606. 28519v1 Announce Type: new Abstract: Training operator-learning models for large-scale problems governed by partial differential equations (PDEs) is challenging due to the curse of dimensionality, memory constraints, and limited training data.