Linearized physics neural networks speed up solving differential equations
Linearized PINN with pretrained nonlinear layers
Machine Learning
Summary
Solving complex physical problems using standard neural networks can be slow and less accurate. The authors propose a new method that first learns a set of building blocks from many example solutions, then quickly solves new problems by mixing these blocks. This approach works faster and with fewer errors than usual methods, especially when there is limited data. It can also handle finer details without retraining.
What this means in practice
- •For computational engineers: Quickly simulate physics problems with fewer computing resources by reusing pretrained neural basis functions.
- •For control system designers: Calibrate or estimate parameters of dynamic systems faster by solving inverse problems efficiently with pretrained model layers.
Authors
Wenhao Chen, Alexandre M. Tartakovsky
Abstract
We propose a linearized Physics-Informed Neural Network (lPINN), a reduced-order neural basis method for forward and inverse differential equations. In an offline stage, lPINN learns operator-compatible continuous neural basis functions from an ensemble of numerical solutions. The basis functions are differentiable through automatic differentiation and are pretrained using solution data together with either derivative information or physics residuals. For each new problem instance, the basis functions are frozen and the solution is obtained by minimizing the governing-equation residual together with applicable initial, boundary, regularization, and observational terms. Unlike surrogate and operator-learning methods, the training data define the trial space offline, while the instance-specific solution is computed online by enforcing the governing physics. Relative to vanilla PINNs, lPINN pretrains the nonlinear hidden-layer representation offline and performs online inference only in the final linear layer. We evaluate lPINN on forward and inverse problems for the advection-diffusion equation, Burgers' equation, and the nonlinear pendulum equation. Compared with vanilla PINNs, lPINN achieves lower solution and parameter errors while reducing online inference times by approximately one to more than three orders of magnitude, with the largest gains generally observed for limited residual or measurement data. Cross-resolution experiments show that the learned continuous representation can be evaluated on finer meshes without retraining and with nearly unchanged accuracy.