PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems

Abstract

Many nonlinear models across physics, chemistry, and biology exhibit multiple solutions for the same parameters, and capturing this entire solution set is essential for understanding pattern-forming systems. Yet existing learned surrogates are fundamentally single-valued: neural operators map each parameter to a single output, and physics-informed neural networks converge to one branch. We develop \textbf{PatternFormer} (PF), a large language model-based framework for learning the multiple solutions of nonlinear partial differential equations. By transforming unordered coexisting solutions into canonical sequences, PF produces structured solution sets in a single autoregressive pass, terminating automatically for finite families and enforcing physical residual constraints for unbounded ones. On nonlinear elliptic problems it recovers all solution branches in one inference step; on Gray--Scott it generates coexisting Turing patterns, including physically valid states absent from the reference data and beyond training. PF can also be sequentially fine-tuned across multistable systems, toward general foundation models for solution landscapes.