Connecting Solutions and Boundary Conditions/Parameters Directly: Solving PDEs in Real Time with PINNs
Keywords: Physics-Informed Neural Networks; Partial Differential Equations; PINNs; PDEs
TL;DR: By establishing the connections between solutions and variable initial conditions, sources and parameters and with offline training, some PDEs are accurately solved with PINNs in real-time for both forward and inverse problems.
Abstract: Physics-Informed Neural Networks (PINNs) have proven to be important tools for solving both forward and inverse problems of partial differential equations (PDEs). However, PINNs face the retraining challenge in which neural networks need to be retrained once the parameters, or boundary/initial conditions change. To address this challenge, meta-learning PINNs train a meta-model across a range of PDE configurations, and the PINN models for new PDE configurations are then generated directly or fine-tuned from the meta-model. Meta-learning PINNs are confronted with either the issue of generalizing to significantly new PDE configurations or the time-consuming process of fine-tuning. By analyzing the mathematical structure of various PDEs, in this paper we establish the direct and mathematically sound connections between PDE solutions and boundary/initial conditions, sources and parameters. The learnable functions in these connections are trained offline in less than 1 hour in most cases. With these connections, the solutions for new PDE configurations can be obtained directly and vice versa, without retraining and fine-tuning at all. Our experimental results indicate that our methods are comparable to vanilla PINNs in terms of accuracy in forward problems, yet at least 400 times faster than them (even over 800 times faster for variable initial/source problems). In inverse problems, our methods are much more accurate than vanilla PINNs while being 80 times faster. Compared with meta-learning PINNs, our methods are much more accurate and about 20 times faster than fine-tuning. Our inference time is less than half a second in forward problems, and at most 3 seconds in inverse problems (less than half a second for variable initial/source problems of linear PDEs). Our code will be made publicly available upon acceptance.
Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)
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Submission Number: 4053
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