arXiv:2609. 16406v1 Announce Type: cross Abstract: Machine learning-based partial differential equations (PDEs) solvers have attracted significant attention in recent years.
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The paper introduces an operator-informed initialization for Fourier Feature Physics-Informed Neural Networks (PINNs) to mitigate spectral bias, where certain frequencies converge slower. By analyzing training dynamics in the Neural Tangent Kernel regime, the authors derive an evolution equation linking convergence rates to the product of the differential operator’s symbol and the spectral density of initialization weights. They propose tailoring the initial weight distribution to the specific PDE, which balances convergence across frequencies and improves prediction accuracy without extra training cost, as confirmed by experiments on linear and nonlinear PDEs.
The paper presents a theoretical framework for certifying the accuracy of physics‑informed neural networks (PINNs) used to solve partial differential equations. It derives generalization bounds that link the residual loss minimized during training to the actual error in the solution space, showing that if the neural approximation stays within a compact subset, a vanishing residual guarantees convergence to the true solution. Deterministic and probabilistic convergence results are provided, offering explicit error guarantees based on residual, boundary, and initial condition errors.
arXiv:2606. 19754v1 Announce Type: new Abstract: Partial differential equations (PDEs) play a central role in modeling complex physical, biological, and engineering systems.
The paper introduces beignet, a neural field architecture that replaces the random Fourier feature embedding in physics-informed neural networks with a trainable multi‑resolution Fourier feature pyramid. By using Fourier interpolation and spectral derivatives, beignet efficiently computes spatial derivatives and scales accuracy through the feature pyramid rather than the neural network size. Experiments show that beignet achieves more accurate PDE solutions with fewer parameters than existing PINN methods and can reach machine‑precision residuals on the inviscid Burgers blowup problem using the Adam optimizer.
The paper explores neural network solvers for infinity and p‑Laplace problems, employing Physics‑Informed Neural Networks (PINNs) and Deep Operator Networks (DeepONets). It addresses computational challenges for large p values (2 to 1000) across 2D and 3D domains, showing advantages over traditional mesh‑based solvers, especially in three dimensions. The authors provide conditional convergence results for PINNs, a universal approximation theorem for DeepONet on the parametric p‑Poisson problem, and validate their methods with numerical experiments comparing performance to conventional approaches.
arXiv:2608. 09494v1 Announce Type: cross Abstract: In this paper we provide Monte Carlo and deep neural network approximations for stochastic representations of solutions to linear elliptic partial differential equations with constant diffusion, drift and killing.