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.
arXiv:2602. 12706v2 Announce Type: replace Abstract: Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs).
arXiv:2604. 07366v2 Announce Type: replace Abstract: Partial differential equations (PDEs) govern nearly every physical process in science and engineering, but solving them at scale remains prohibitively expensive.
The paper introduces Neptune, a method that uses independent coordinate neural networks to infer parameter fields in multi-physics PDEs from sparse measurements. Neptune can accurately estimate parameters with nonlinear, spatiotemporal variations, outperforming existing techniques by reducing estimation errors by up to two orders of magnitude and improving dynamic response predictions by a factor of ten. It also demonstrates strong physical extrapolation, enabling reliable predictions beyond the training data.
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: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 introduces the Local Gradient Neural Operator (LGNO), a lightweight and interpretable neural operator designed for field temporal evolution prediction and source identification in mechanical problems. LGNO leverages nonlinear gradient discretization priors and multilayer perceptron convolutional layers to learn translation‑invariant local kernels resembling discrete stencils, with a zero‑consistent stencil factorization that separates coefficient learning from field reconstruction. Experiments on a range of PDE benchmarks—including linear, nonlinear, static, dynamic, low‑ and high‑dimensional cases—demonstrate that LGNO achieves comparable accuracy to global neural operators while using fewer parameters and maintaining rollout stability across diffusion, flow, and quantum phenomena.
Field temporal prediction and source identification constitute canonical problems in dynamical systems. Conventional approaches to these problems depend on a thorough understanding of the governing pa...
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...
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.
The paper introduces two multi-stage neural operator learning frameworks—Deep Collocation Neural Operator (DCNO) and Deep Galerkin Neural Operator (DGNO)—for efficiently computing convolution integrals. DCNO is a supervised method that iteratively refines operator approximations by learning residuals from data pairs, while DGNO is an unsupervised approach that uses the weak form of a PDE residual when the operator can be represented by a PDE. Both frameworks build basis operators across multiple training stages, yielding markedly higher accuracy than one-shot learning and achieving near machine‑precision results for convolution problems, with significant efficiency gains for repeated queries or parametric variations.
arXiv:2607. 28762v1 Announce Type: new Abstract: This work embeds feature interaction modules derived from factorization machines (FMs) into physics-informed neural networks (PINNs) and neural operator learning, to enhance model expressiveness for solution manifolds of parameterized partial differential equations (PDEs).
arXiv:2605. 25413v3 Announce Type: replace-cross Abstract: Neural operators learn mappings from function-dependent inputs to solutions, providing an effective framework for solving partial differential equations (PDEs).