Physics-informed fine-tuning of foundation models for partial differential equations

Pretrained partial differential equation (PDE) foundation models can generalize across different equations, but adapting them to unseen PDE systems typically requires dense solution data, which is often expensive or unavailable. To address this limitation, we propose an unsupervised PDE-based finetuning framework that eliminates the need for ground-truth solutions. We first pretrain a neighborhood attention Transformer on diverse time-dependent PDEs spanning varying spatial scales, yielding transferable representations across heterogeneous equations. In the adaptation stage, we construct a physics-based objective using the PDE residual and boundary conditions, and finetune the model on unseen equations via low-rank adaptation (LoRA). To address the uneven learning across physical quantities in standard LoRA, we introduce NSLoRA, a Newton-Schulz orthogonalized variant that rebalances adaptation. Our method achieves performance comparable to supervised LoRA finetuning without requiring any ground-truth solutions, while consistently outperforming competitive neural operator baselines and recent PDE foundation models across heterogeneous PDE benchmarks spanning multiple spatial dimensions.

Scientific foundation models (SciFMs) aim to learn generalizable representations of physical systems governed by partial differential equations (PDEs), enabling transfer across tasks and domains. While physics-informed methods, which leverage PDE residuals as supervisory signals, have shown promise in scientific machine learning (SciML) for improving accuracy and reducing data requirements, their potential in the context of SciFMs remains relatively unexplored. In this evaluation study, we investigate whether (and how) physics-informed pre-training improves the generalization, robustness, and data efficiency of SciFMs. We conduct systematic experiments across a diverse set of PDEs, ranging from simple problems with periodic boundary conditions to more challenging systems such as the Navier-Stokes equations and non-periodic geometries. Our results show that physics-informed pre-training provides clear benefits in nice,'' e.g., structured, well-aligned settings: it enhances generalization and reduces data dependence, compared to data-only pre-training. However, these advantages diminish significantly as the downstream tasks become harder,'' e.g., as they involve discontinuities or deviate from the pre-training distribution. In complex or structurally different problems, such as those involving new boundary conditions or PDE operators, physics-informed models may perform only on par with---or even worse---than data-driven baselines. While residual-based pre-training helps in idealized regimes, realizing broadly transferable SciFMs will likely require subtler spatiotemporal inductive biases and more principled integration of physical knowledge into model architectures.

Physics-informed neural networks (PINNs) have garnered significant interest for their potential in solving partial differential equations (PDEs) that govern a wide range of physical phenomena. By incorporating physical laws into the learning process, PINN models have demonstrated the ability to learn physical outcomes reasonably well. However, current PINN approaches struggle to predict or solve new PDEs effectively when there is a lack of training examples, indicating they do not generalize well to unseen problem instances. In this paper, we present a transferable learning approach for PINNs premised on a fast Pseudoinverse PINN framework (Pi-PINN). Pi-PINN learns a transferable physics-informed representation in a shared embedding space and enables rapid solving of both known and unknown PDE instances via closed-form head adaptation using a least-squares-optimal pseudoinverse under PDE constraints. We further investigate the synergies between data-driven multi-task learning loss and physics-informed loss, providing insights into the design of more performant PINNs. We demonstrate the effectiveness of Pi-PINN on various PDE problems, including Poisson's equation, Helmholtz equation, and Burgers' equation, achieving fast and accurate physics-informed solutions without requiring any data for unseen instances. Pi-PINN can produce predictions 100-1000 times faster than a typical PINN, while producing predictions with 10-100 times lower relative error than a typical data-driven model even with only two training samples. Overall, our findings highlight the potential of transferable representations with closed-form head adaptation to enhance the efficiency and generalization of PINNs across PDE families and scientific and engineering applications.