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Adaptive Correction for Ensuring Conservation Laws in Neural Operators
Christopher Budd1, Chaoyu Liu2, Carola-Bibiane Schoenlieb2
1: University of Bath, United Kingdom; 2: University of Cambridge, United Kingdom
Physical laws, such as the conversation of mass and momentum, are fundamental principles in many physical systems. Neural operators have achieved promising performance in learning the solutions to those systems, but often fail to ensure conservation. Existing methods typically enforce strict conservation via hand-crafted post-processing or architectural constraints, leading to limited model flexibility and adaptability. In this talk I will present a novel plug-and-play adaptive correction approach to ensure the conservation of fundamental linear and quadratic quantities for neural operator outputs. The method introduces a lightweight learnable operator to adap- tively enforce the target conservation law during training. This method allows the model to flex- ibly and adaptively correct its output to guarantee strict conservation. I will provide a theoretical result showing that the correction method does not hamper the expression ability of neural operators and can potentially achieve lower recontruction loss than their conservation-constrained counterparts. The method will be demonstrated across muliple neural operator architectures and represen- tative PDEs. Extensive experiments will show tha incorporating the correction method into baseline models significantly improves both accuracy and stability. In addition, the experimental results will demonstrate that this approach consistently achieves superior performance over widely used conservation-enforcement techniques on various PDE benchmarks.
