Late Fusion Neural Operators for Parameterized Partial Differential Equations

Eva van Tegelen, Taniya Kapoor, George A. K. van Voorn, Ioannis N. Athanasiadis

Keywords: Neural operators, PDEs

Abstract: Developing models that can accurately predict PDE behaviour for unseen parameter values is crucial for achieving robust and reliable generalization in scientific and engineering applications. In practice, variations in physical parameters induce distribution shifts between training and prediction regimes, making extrapolation a central challenge. As a result, the way parameters are incorporated into neural operator models plays a key role in their ability to generalize, particularly when state and parameter representations are entangled. In this work, we introduce a novel architecture for parameterized PDEs, that employs a late fusion neural operator architecture to improve performance inside and outside the training distribution. Our approach combines neural operators for learning latent state representations with sparse regression to incorporate parameter structure in an interpretable manner. By explicitly separating the learning of state dynamics from parameter dependence, the proposed Late Fusion Operator enables controlled and interpretable incorporation of parameter effects. Across a range of benchmark PDEs, the proposed Late Fusion Operator consistently outperforms existing approaches (best-performing in approximately 75% of experiments, with 32–86% improvement compared to the second-best method), demonstrating strong generalization across both in-domain and out-of-domain parameter regimes.

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Submission Number: 12