PDEformer-2 Extends Foundation Models to Two-Dimensional PDEs

Scientific simulators are often built one equation family at a time: a numerical discretization is selected for a particular domain and boundary condition, while learned operators are commonly trained for a single distribution of PDE instances. That specialization becomes costly when the equation itself, its coefficients, the geometry, or the number of dependent variables changes. PDEformer-2 proposes a different interface: represent the PDE as a computational graph, condition a shared neural solver on that graph, and query the resulting solution at arbitrary space-time coordinates.

The paper extends the authors' earlier one-dimensional PDEformer work to two-dimensional problems. It is presented as a differentiable model that can generate approximate solutions quickly across a broad set of PDE specifications. For new unseen PDEs, the authors adapt the pretrained model using a limited number of examples.

Core Contribution

The central contribution is a single model that treats a symbolic PDE specification as an input rather than as an architectural choice made before training. The computational graph encodes the PDE form and is designed to accommodate variation in domains, boundary conditions, time dependence, and numbers of variables. The model therefore receives a representation of the equation it is being asked to solve, rather than relying on a separately trained model for each PDE family.

The output is mesh-free: PDEformer-2 predicts the solution at queried spatial and temporal coordinates rather than only on a fixed output grid. This coordinate-query interface also preserves differentiability from the PDE description and parameters to predicted solution values, supporting the inverse-problem experiments considered in the paper.

Technical Approach

PDEformer-2 combines graph-based conditioning with coordinate-based solution prediction. The graph representation supplies the symbolic structure of the governing equation, while the query coordinates specify where the solution is evaluated. Pretraining uses a diverse 40 TB dataset of two-dimensional PDE data, spanning variation in symbolic forms and problem settings.

The model is evaluated in two settings. For PDEs resembling the pretraining distribution, the authors consider zero-shot prediction. For unseen equations, they adapt the foundation model with a small number of examples. The paper also applies the differentiable model to inverse problems, recovering coefficient scalars and coefficient fields from observations.

Results and Analysis

The reported experiments support the narrower claim that pretraining can improve data efficiency for related two-dimensional PDE tasks. In low-data adaptation, the paper reports faster learning than several specialized models and smaller errors when fewer than 100 samples are available. This makes the approach relevant where generating new task-specific simulation data is limited or costly.

The zero-shot results are explicitly scoped to PDEs that resemble the pretraining distribution. A computational graph can express many equations, but representing a PDE is not by itself evidence that the model has learned its solution behavior. The results therefore support PDEformer-2 as a reusable model for represented two-dimensional problem families rather than as evidence of uniform generalization to arbitrary PDEs.

The inverse-problem results show that the model's differentiable predictions can be used to recover coefficient scalars and fields in the reported experiments. The paper characterizes these reconstructions as reasonable, rather than presenting a broad comparison across inverse-PDE methods. The strongest practical reading is that the model is most applicable when the target equation is near its training support and limited adaptation data are available.

Limits in Practice

The evaluation is confined to two-dimensional PDEs, and zero-shot accuracy is explicitly distribution-dependent. The 40 TB training dataset is substantial, but its scale alone does not establish coverage of rare operators, extreme coefficient regimes, irregular geometries, or long-time dynamics. The inverse experiments demonstrate selected coefficient-recovery cases rather than a comparison across noise levels and identifiability conditions. These boundaries make the empirical case substantial but narrower than the phrase “versatile solver” can imply.