Unisolver: PDEConditional Transformers Are Universal PDE Solvers
Abstract
Deep models have recently emerged as a promising tool to solve partial differential equations (PDEs), known as neural PDE solvers. While neural solvers trained from either simulation data or physicsinformed loss can solve the PDEs reasonably well, they are mainly restricted to a specific set of PDEs, e.g. a certain equation or a finite set of coefficients. This bottleneck limits the generalizability of neural solvers, which is widely recognized as its major advantage over numerical solvers. In this paper, we present the Universal PDE solver (Unisolver) capable of solving a wide scope of PDEs by leveraging a Transformer pretrained on diverse data and conditioned on diverse PDEs. Instead of simply scaling up data and parameters, Unisolver stems from the theoretical analysis of the PDEsolving process. Our key finding is that a PDE solution is fundamentally under the control of a series of PDE components, e.g. equation symbols, coefficients, and initial and boundary conditions. Inspired by the mathematical structure of PDEs, we define a complete set of PDE components and correspondingly embed them as domainwise (e.g. equation symbols) and pointwise (e.g. boundaries) conditions for Transformer PDE solvers. Integrating physical insights with recent Transformer advances, Unisolver achieves consistent stateoftheart results on three challenging largescale benchmarks, showing impressive gains and endowing favorable generalizability and scalability.
 Publication:

arXiv eprints
 Pub Date:
 May 2024
 DOI:
 10.48550/arXiv.2405.17527
 arXiv:
 arXiv:2405.17527
 Bibcode:
 2024arXiv240517527Z
 Keywords:

 Computer Science  Machine Learning;
 Computer Science  Artificial Intelligence;
 Mathematics  Numerical Analysis