Reduced-order modeling for parameterized PDEs via implicit neural representations

TL;DR

This paper introduces a novel data-driven reduced-order modeling approach for parametrized PDEs using implicit neural representations and neural ODEs, enabling efficient and accurate many-query solutions with physics-informed fine-tuning.

Contribution

The work develops a PDE encoding framework with a hypernetwork-inferrable neural ODE and physics-informed loss, improving efficiency and accuracy for parametrized PDE solutions.

Findings

Achieves up to 1000x speedup in solving PDEs.

Maintains approximately 1% relative error compared to ground truth.

Enables unsupervised fine-tuning on unseen parameters.

Abstract

We present a new data-driven reduced-order modeling approach to efficiently solve parametrized partial differential equations (PDEs) for many-query problems. This work is inspired by the concept of implicit neural representation (INR), which models physics signals in a continuous manner and independent of spatial/temporal discretization. The proposed framework encodes PDE and utilizes a parametrized neural ODE (PNODE) to learn latent dynamics characterized by multiple PDE parameters. PNODE can be inferred by a hypernetwork to reduce the potential difficulties in learning PNODE due to a complex multilayer perceptron (MLP). The framework uses an INR to decode the latent dynamics and reconstruct accurate PDE solutions. Further, a physics-informed loss is also introduced to correct the prediction of unseen parameter instances. Incorporating the physics-informed loss also enables the model to be fine-tuned in an unsupervised manner on unseen PDE parameters. A numerical experiment is performed on a two-dimensional Burgers equation with a large variation of PDE parameters. We evaluate the proposed method at a large Reynolds number and obtain up to speedup of O(10^3) and ~1% relative error to the ground truth values.