How to Select Physics-Informed Neural Networks in the Absence of Ground Truth: A Pareto Front-Based Strategy
Keywords: Machine Learning, Physics-Informed Neural Networks, Partial Differential Equation, Mesh-free Simulation
TL;DR: We outline a Pareto front-based strategy to select more accurate Physics-informed NNs when they are used as a mesh-free alternative to modelling and no ground truth results are available by definition.
Abstract: Physics-informed neural networks (PINNs) have been proposed as a potential route to inverse modelling or mesh-free alternative to numerical methods for partial differential equations (PDEs). However, these problems typically lack ground truth, making selection of more accurate PINN models difficult, especially with processes such as hyper-parameter tuning. This is exacerbated as PINNs need to balance multiple objectives, comprising the governing PDEs, associated boundary/initial conditions, and/or point data. Under this multi-objective optimization framework, the ideal PINN solution is one that achieves zero loss across all components although this is not typical, resulting in a Pareto set of models. Nonetheless, there are objectively-preferred models based on congruence to unknown ground truth. In this context, we propose a Pareto front-based analysis to help identify better performing models. First, an approximation to the Pareto set of solutions with minimal PINN loss is constructed for different balances of loss weights. A loss weight located on the convex bulge of the Pareto front is then selected to rescale the training loss across all solutions. Across our experiments, this rescaling demonstrates a strong correlation between the rescaled PINN loss and mean squared error (MSE) relative to simulated ground truth, thereby illustrating potential PINN model selection.
Submission Number: 81
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