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This paper introduces PDEformer, a neural solver for partial differential equations (PDEs) capable of simultaneously addressing various types of PDEs. We propose to represent the PDE in the form of a computational graph, facilitating the…
Bridging physics and deep learning is a topical challenge. While deep learning frameworks open avenues in physical science, the design of physically-consistent deep neural network architectures is an open issue. In the spirit of…
Solving partial differential equations (PDEs) with machine learning typically requires training a new neural network for every new equation. This optimization is slow. We introduce MetaColloc. It is an optimization-free and data-free…
Partial differential equations (PDEs) that fit scientific data can represent physical laws with explainable mechanisms for various mathematically-oriented subjects, such as physics and finance. The data-driven discovery of PDEs from…
Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often…
Deriving governing equations in Electromagnetic (EM) environment based on first principles can be quite tough when there are some unknown sources of noise and other uncertainties in the system. For nonlinear multiple-physics electromagnetic…
A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time discretised version of the invariant is…
Physics-Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding physical laws into neural network training. However, traditional PINN models are typically designed…
Stochastic branching algorithms provide a useful alternative to grid-based schemes for the numerical solution of partial differential equations, particularly in high-dimensional settings. However, they require a strict control of the…
Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…
In this work, we present an adjoint-based method for discovering the underlying governing partial differential equations (PDEs) given data. The idea is to consider a parameterized PDE in a general form and formulate a PDE-constrained…
Partial differential equations (PDEs) govern physical phenomena across the full range of scientific scales, yet their computational solution remains one of the defining challenges of modern science. This critical review examines two mature…
We present PDE-FM, a modular foundation model for physics-informed machine learning that unifies spatial, spectral, and temporal reasoning across heterogeneous partial differential equation (PDE) systems. PDE-FM combines spatial-spectral…
Within recent years, considerable progress has been made regarding high-performance solvers for Partial Differential Equations (PDEs), yielding potential gains in efficiency compared to industry standard tools. However, the latter largely…
We give a systematic method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same method, applied to PDEs with constant dissipative structure,…
In this paper, we develop a class of interacting particle Langevin algorithms to solve inverse problems for partial differential equations (PDEs). In particular, we leverage the statistical finite elements (statFEM) formulation to obtain a…
The propagation of electromagnetic waves in general media is modeled by the time-dependent Maxwell's partial differential equations (PDEs), coupled with constitutive laws that describe the response of the media. In this work, we focus on…
Partial differential equations (PDEs) are fundamental for modeling complex natural and physical phenomena. In many real-world applications, however, observational data are extremely sparse, which severely limits the applicability of both…
Modeling physical systems in a generative manner offers several advantages, including the ability to handle partial observations, generate diverse solutions, and address both forward and inverse problems. Recently, diffusion models have…
Identifying governing partial differential equations (PDEs) from noisy spatiotemporal data remains challenging due to differentiation-induced noise amplification and ambiguity from overcomplete libraries. We propose a prior-informed…