Physics Informed Symbolic Networks
This work addresses the problem of efficiently solving PDEs for researchers in computational physics and engineering, offering a novel hybrid approach that is incremental but with significant performance improvements.
The paper tackles solving partial differential equations (PDEs) by introducing Physics Informed Symbolic Networks (PISN) that use physics-informed loss to derive symbolic solutions, achieving performance comparable to PINNs on benchmarks like Kovasznay flow and viscous Burger's equations, and Physics-informed Neurosymbolic Networks (PINSN) that combine symbolic networks with MLPs to yield a 2-3 orders of magnitude performance gain over standard PINNs.
We introduce Physics Informed Symbolic Networks (PISN) which utilize physics-informed loss to obtain a symbolic solution for a system of Partial Differential Equations (PDE). Given a context-free grammar to describe the language of symbolic expressions, we propose to use weighted sum as continuous approximation for selection of a production rule. We use this approximation to define multilayer symbolic networks. We consider Kovasznay flow (Navier-Stokes) and two-dimensional viscous Burger's equations to illustrate that PISN are able to provide a performance comparable to PINNs across various start-of-the-art advances: multiple outputs and governing equations, domain-decomposition, hypernetworks. Furthermore, we propose Physics-informed Neurosymbolic Networks (PINSN) which employ a multilayer perceptron (MLP) operator to model the residue of symbolic networks. PINSNs are observed to give 2-3 orders of performance gain over standard PINN.