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. iFOL employs a purely physics-informed loss function derived via the Method of Weighted Residuals. The predicted neural field serves as the test function, resulting in the backpropagation of discrete residuals during the PDE encoding and decoding stages.
Compared to the state-of-the-art neural operators, iFOL introduces several key innovations: (1) it bypasses the costly multi-network and supervised encode–process–decode pipeline of conditional neural fields for parametric PDEs; (2) it yields accurate parametric fields and solution-to-parameter gradients, enabling efficient sensitivity analysis regardless of response count; (3) it effectively captures sharp solution discontinuities, which are often challenging for some neural operator models; and (4) it is mesh and geometry agnostic, enabling zero-shot generalization to arbitrary domains. We critically assess these features and analyze the network’s ability to generalize to unseen samples across both stationary and transient PDEs. The method is also compared against baseline operator-learning approaches, demonstrating its potential for tackling complex problems in computational mechanics.
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