  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.
