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 physics-informed 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 pre-trained on diverse data and conditioned on diverse PDEs. Instead of simply scaling up data and parameters, Unisolver stems from the theoretical analysis of the PDE-solving 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 domain-wise (e.g. equation symbols) and point-wise (e.g. boundaries) conditions for Transformer PDE solvers. Integrating physical insights with recent Transformer advances, Unisolver achieves consistent state-of-the-art results on three challenging large-scale benchmarks, showing impressive gains and endowing favorable generalizability and scalability.

翻译：深度模型近年来作为求解偏微分方程（PDEs）的有力工具崭露头角，被称为神经PDE求解器。尽管基于仿真数据或物理信息损失训练的神经求解器能够较好地求解PDE，但其通常局限于特定PDE集合（例如单一方程或有限系数集）。这一瓶颈限制了神经求解器的泛化能力，而泛化能力正是其相较于传统数值求解器被广泛认可的核心优势。本文提出通用PDE求解器（Unisolver），通过利用基于多样化数据预训练并受多样化PDE条件约束的Transformer，实现了对广泛PDE类型的高效求解。Unisolver并非简单扩展数据与参数规模，而是源于对PDE求解过程的理论分析。我们的核心发现是：PDE解本质上受一系列PDE要素控制，例如方程符号、系数以及初始和边界条件。受PDE数学结构的启发，我们定义了完整的PDE要素集合，并相应地将它们嵌入为Transformer PDE求解器的域级（如方程符号）与点级（如边界）条件。通过将物理洞见与Transformer前沿技术相结合，Unisolver在三个具有挑战性的大规模基准测试中取得了持续领先的性能，展现出显著的性能提升，并赋予模型优异的泛化能力与可扩展性。