AI 中文总结
该研究提出基于大语言模型的PatternFormer框架，可学习非线性偏微分方程的多种解，在非线性椭圆问题和Gray-Scott问题上表现优异，还可微调以构建解景观通用基础模型。
AI 中文摘要
物理、化学和生物学领域的许多非线性模型在相同参数下会呈现多种解，捕获整个解集对理解模式形成系统至关重要。然而现有的学习代理本质上是单值的：神经算子将每个参数映射到单个输出，物理信息神经网络收敛到一个分支。我们开发了PatternFormer（PF），一种基于大语言模型的框架，用于学习非线性偏微分方程的多种解。通过将无序共存的解转换为规范序列，PF在单次自回归传递中生成结构化解集，对有限族自动终止，对无限族则施加物理残差约束。在非线性椭圆问题上，它在一次推理步骤中恢复所有解分支；在Gray-Scott问题上，它生成共存的图灵模式，包括参考数据中不存在且超出训练范围的物理有效状态。PF还可在多稳态系统上进行顺序微调，朝着构建解景观的通用基础模型发展。
英文摘要
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.
