Reduced-order modeling for parameterized PDEs via implicit neural representations
This work addresses computational efficiency for PDE simulations in engineering or physics, but it appears incremental as it builds on existing implicit neural representation and neural ODE methods.
The authors tackled the problem of efficiently solving parameterized partial differential equations (PDEs) for many-query problems by developing a data-driven reduced-order modeling approach using implicit neural representations, achieving up to O(10^3) speedup and ~1% relative error in a numerical experiment on a two-dimensional Burgers equation.
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.