Transformers as Neural Operators for Solutions of Differential Equations with Finite Regularity

Neural operator learning models have emerged as very effective surrogates in data-driven methods for partial differential equations (PDEs) across different applications from computational science and engineering. Such operator learning models not only predict particular instances of a physical or biological system in real-time but also forecast classes of solutions corresponding to a distribution of initial and boundary conditions or forcing terms. % DeepONet is the first neural operator model and has been tested extensively for a broad class of solutions, including Riemann problems. Transformers have not been used in that capacity, and specifically, they have not been tested for solutions of PDEs with low regularity. % In this work, we first establish the theoretical groundwork that transformers possess the universal approximation property as operator learning models. We then apply transformers to forecast solutions of diverse dynamical systems with solutions of finite regularity for a plurality of initial conditions and forcing terms. In particular, we consider three examples: the Izhikevich neuron model, the tempered fractional-order Leaky Integrate-and-Fire (LIF) model, and the one-dimensional Euler equation Riemann problem. For the latter problem, we also compare with variants of DeepONet, and we find that transformers outperform DeepONet in accuracy but they are computationally more expensive.

RiemannONets: Interpretable Neural Operators for Riemann Problems

Developing the proper representations for simulating high-speed flows with strong shock waves, rarefactions, and contact discontinuities has been a long-standing question in numerical analysis. Herein, we employ neural operators to solve Riemann problems encountered in compressible flows for extreme pressure jumps (up to $10^{10}$ pressure ratio). In particular, we first consider the DeepONet that we train in a two-stage process, following the recent work of Lee and Shin, wherein the first stage, a basis is extracted from the trunk net, which is orthonormalized and subsequently is used in the second stage in training the branch net. This simple modification of DeepONet has a profound effect on its accuracy, efficiency, and robustness and leads to very accurate solutions to Riemann problems compared to the vanilla version. It also enables us to interpret the results physically as the hierarchical data-driven produced basis reflects all the flow features that would otherwise be introduced using ad hoc feature expansion layers. We also compare the results with another neural operator based on the U-Net for low, intermediate, and very high-pressure ratios that are very accurate for Riemann problems, especially for large pressure ratios, due to their multiscale nature but computationally more expensive. Overall, our study demonstrates that simple neural network architectures, if properly pre-trained, can achieve very accurate solutions of Riemann problems for real-time forecasting.

Deep neural operators can serve as accurate surrogates for shape optimization: A case study for airfoils

Deep neural operators, such as DeepONets, have changed the paradigm in high-dimensional nonlinear regression from function regression to (differential) operator regression, paving the way for significant changes in computational engineering applications. Here, we investigate the use of DeepONets to infer flow fields around unseen airfoils with the aim of shape optimization, an important design problem in aerodynamics that typically taxes computational resources heavily. We present results which display little to no degradation in prediction accuracy, while reducing the online optimization cost by orders of magnitude. We consider NACA airfoils as a test case for our proposed approach, as their shape can be easily defined by the four-digit parametrization. We successfully optimize the constrained NACA four-digit problem with respect to maximizing the lift-to-drag ratio and validate all results by comparing them to a high-order CFD solver. We find that DeepONets have low generalization error, making them ideal for generating solutions of unseen shapes. Specifically, pressure, density, and velocity fields are accurately inferred at a fraction of a second, hence enabling the use of general objective functions beyond the maximization of the lift-to-drag ratio considered in the current work.