A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries
Reza Najian Asl, Yusuke Yamazaki, Kianoosh Taghikhani, Mayu Muramatsu, Markus Apel, Shahed Rezaei
TL;DR
The paper introduces iFOL, a physics-informed, geometry-agnostic framework that learns continuous parametric solutions to PDEs on arbitrary geometries by fusing implicit neural representations, FiLM conditioning, and second-order meta-learning. It replaces the traditional encode-process-decode pipeline with a PDE-encoded, physics-guided loss that yields accurate solution fields and Jacobians without labeled data. The approach is validated on stationary and transient problems across hyperelasticity, diffusion, and phase-field equations, demonstrating zero-shot super-resolution and competitive computational efficiency. The work offers a versatile tool for parametric studies and gradient-based optimization in computational mechanics, with code openly available.
Abstract
In this work, we introduce implicit Finite Operator Learning (iFOL) for the continuous and parametric solution of partial differential equations (PDEs) on arbitrary geometries. We propose a physics-informed encoder-decoder network to establish the mapping between continuous parameter and solution spaces. The decoder constructs the parametric solution field by leveraging an implicit neural field network conditioned on a latent or feature code. Instance-specific codes are derived through a PDE encoding process based on the second-order meta-learning technique. In training and inference, a physics-informed loss function is minimized during the PDE encoding and decoding. iFOL expresses the loss function in an energy or weighted residual form and evaluates it using discrete residuals derived from standard numerical PDE methods. This approach results in the backpropagation of discrete residuals during both training and inference. iFOL features several key properties: (1) its unique loss formulation eliminates the need for the conventional encode-process-decode pipeline previously used in operator learning with conditional neural fields for PDEs; (2) it not only provides accurate parametric and continuous fields but also delivers solution-to-parameter gradients without requiring additional loss terms or sensitivity analysis; (3) it can effectively capture sharp discontinuities in the solution; and (4) it removes constraints on the geometry and mesh, making it applicable to arbitrary geometries and spatial sampling (zero-shot super-resolution capability). We critically assess these features and analyze the network's ability to generalize to unseen samples across both stationary and transient PDEs. The overall performance of the proposed method is promising, demonstrating its applicability to a range of challenging problems in computational mechanics.