Physics-Informed Deep B-Spline Networks
Keywords: Physics-informed machine learning, B-splines, Partial differential equations (PDEs)
Abstract: Physics-informed machine learning provides an approach to combing data and governing physics laws for solving complex partial differential equations (PDEs). However, efficiently solving PDEs with varying parameters and changing initial conditions and boundary conditions (ICBCs) remains an open challenge. We propose a hybrid framework that uses a neural network to learn B-spline control points to approximate solutions to PDEs with varying system and ICBC parameters. The proposed network can be trained efficiently as one can directly specify ICBCs without imposing losses, calculate physics-informed loss functions through analytical formulas, and requires only learning the weights of B-spline functions as opposed to both weights and basis as in traditional neural operator learning methods. We show theoretical guarantees that the proposed B-spline networks are universal approximators of arbitrary dimensional PDEs under certain conditions. We also demonstrate in experiments that the proposed B-spline network can solve problems with discontinuous ICBCs and outperforms existing methods, and is able to learn solutions of 3D heat equations with diverse initial conditions.
Primary Area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
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Submission Number: 12276
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