Hypernetwork-based Meta-Learning for Low-Rank Physics-Informed Neural Networks

In various engineering and applied science applications, repetitive numerical simulations of partial differential equations (PDEs) for varying input parameters are often required (e.g., aircraft shape optimization over many design parameters) and solvers are required to perform rapid execution. In this study, we suggest a path that potentially opens up a possibility for physics-informed neural networks (PINNs), emerging deep-learning-based solvers, to be considered as one such solver. Although PINNs have pioneered a proper integration of deep-learning and scientific computing, they require repetitive time-consuming training of neural networks, which is not suitable for many-query scenarios. To address this issue, we propose a lightweight low-rank PINNs containing only hundreds of model parameters and an associated hypernetwork-based meta-learning algorithm, which allows efficient approximation of solutions of PDEs for varying ranges of PDE input parameters. Moreover, we show that the proposed method is effective in overcoming a challenging issue, known as "failure modes" of PINNs.

A range characterization of the single-quadrant ADRT

This work characterizes the range of the single-quadrant approximate discrete Radon transform (ADRT) of square images. The characterization is given in the form of linear constraints that ensure the exact and fast inversion formula [Rim, Appl. Math. Lett. 102 106159, 2020] yields a square image in a stable manner. The range characterization is obtained by first showing that the transform is a bijection between images supported on infinite half-strips, then identifying the linear subspaces that stay finitely supported under the inversion formula.

Depth separation for reduced deep networks in nonlinear model reduction: Distilling shock waves in nonlinear hyperbolic problems

Classical reduced models are low-rank approximations using a fixed basis designed to achieve dimensionality reduction of large-scale systems. In this work, we introduce reduced deep networks, a generalization of classical reduced models formulated as deep neural networks. We prove depth separation results showing that reduced deep networks approximate solutions of parametrized hyperbolic partial differential equations with approximation error $\epsilon$ with $\mathcal{O}(|\log(\epsilon)|)$ degrees of freedom, even in the nonlinear setting where solutions exhibit shock waves. We also show that classical reduced models achieve exponentially worse approximation rates by establishing lower bounds on the relevant Kolmogorov $N$-widths.

Exact and Fast Inversion of the Approximate Discrete Radon Transform

We give an exact inversion formula for the approximate discrete Radon transform introduced in [Brady, SIAM J. Comput., 27(1), 107--119] that is of cost $O(N \log N)$ for a square 2D image with N pixels.