Papers for
software developers for physics simulation
Papers whose findings have a practical use for this group, as judged from the abstract. Open a paper to read what it means in practice.
Adaptive randomized neural networks improve solving complex pdes efficiently
Adaptive Randomized Neural Networks for PDEs: Structure-Aware Feature Redistribution and Residual-Driven Block Enrichment
Abstract: Randomized neural networks retain an output-linear least-squares solve, but globally sampled random features may fail to capture the spatial and scale structure of PDE solutions. We propose SFRBE--RaNN, a two-stage adaptive trial-space construction. SFR redistributes a fixed feature budget according to regional indicator masses and adapts regional weight scales through low-dimensional residual-based optimization. RBE constructs nonredundant enrichments by screening multiple activation families, removing represented response directions, and selecting the smallest admissible prefix. For solution-dependent operators, candidates are screened with a frozen operator and accepted only after nonlinear residual verification. For fixed linear problems, we establish an exact discrete residual-reduction identity and a conditional solution-error estimate. Building on this identity, we combine safeguarded SFR with RBE capture factors to derive a finite-horizon error bound involving baseline best approximation and empirical consistency, together with a conditional random-search success-probability bound. Numerical experiments on localized, multiscale, and nonlinear space--time problems demonstrate the method's effectiveness.
Pde solvers benefit from machine learning trained on smaller problems
Warm-starting PDE solvers with any-dimensional machine learning
Abstract: Any-dimensional machine learning models, such as graph neural networks (GNNs), can be naturally trained and evaluated on inputs of different sizes and dimensions. Inspired by the GNN transferability literature, we show mathematical conditions under which a partial differential equation (PDE) learning-based solver can be trained in small dimensions and directly applied to solve a higher dimensional PDE in a zero-shot fashion. These conditions are based on symmetries in both the partial differential equation and the initial data. When the equations satisfy the symmetries but the data does not, which is the case for many PDEs arising from physics, we show that our theory gives a principled way of warm-starting low-dimensional PDE solvers for higher dimensional PDEs. We apply this method on the heat equation, Burgers' equation, and the compressible Navier--Stokes equations, improving the performance in both zero-shot and typical training regimes on high dimensional data. For example, we train a surrogate model on 2D Navier--Stokes data and achieve better results on 3D test data than a baseline surrogate model trained on 3D data, while only using 12$\%$ of the flops and 20$\%$ of the total data size.