Multifidelity modeling for Physics-Informed Neural Networks (PINNs)

Highlights: • Low-rank multifidelity construction of surrogate models for Physics-Informed Neural Networks. • Width, depth, and optimization hyperparameters of Physics-Informed Neural Networks as a form of fidelity. • Theoretical and implementation considerations to guide tuning and set up. • Cost and accuracy improvement on four canonical forward partial differential equation problems. Multifidelity simulation methodologies are often used in an attempt to judiciously combine low-fidelity and high-fidelity simulation results in an accuracy-increasing, cost-saving way. Candidates for this approach are simulation methodologies for which there are fidelity differences connected with significant computational cost differences. Physics-informed Neural Networks (PINNs) are candidates for these types of approaches due to the significant difference in training times required when different fidelities (expressed in terms of architecture width and depth as well as optimization criteria) are employed. In this paper, we propose a particular multifidelity approach applied to PINNs that exploits low-rank structure. We demonstrate that width, depth, and optimization criteria can be used as parameters related to model fidelity and show numerical justification of cost differences in training due to fidelity parameter choices. We test our multifidelity scheme on various canonical forward PDE models that have been presented in the emerging PINNs literature.