Hybrid Numerical PINNs: On the effectiveness of numerical differentiation for non-analytic problems
Keywords: Scientific Machine Learning, Physics-Informed Neural Networks, Automatic Differentiation, Partial Differential Equations
Abstract: This work demonstrates that automatic differentiation has strong limitations when employed to compute physical derivatives in a general physics-informed framework, therefore limiting the range of applications that these methods can address. A hybrid approach is proposed, combining deep learning and traditional numerical solvers such as the finite element method, to address the shortcomings of automatic differentiation. This novel approach enables the exact imposition of Dirichlet boundary conditions in a seamless manner, and more complex, non analytical problems can be solved. Finally, enriched inputs can be used by the model to help convergence. The proposed approach is flexible and can be incorporated into any physics-informed model. Our hybrid gradient computation proposal is also up to two orders of magnitude faster than automatic differentiation, as its numerical cost is independent of the complexity of the trained model. Several numerical applications are provided to illustrate the discussion.
Primary Area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
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Submission Number: 9786
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