QUANTUM FOURIER NETWORKS FOR SOLVING PARA-METRIC PDES

Abstract

Many real-world problems like modelling environment dynamics, physical processes, time series etc., involve solving Partial Differential Equations (PDEs) parameterized by problem-specific conditions. Recently, a deep learning architecture called Fourier Neural Operator (FNO) proved to be capable of learning solutions of given PDE families, for any initial conditions as input. Given the advancements in quantum hardware and the recent results in quantum machine learning methods, we propose three quantum circuits, inspired by the FNO, to learn this functional mapping for PDEs. The proposed algorithms are distinguished based on the trade-off between depth and their similarity to the classical FNO. At their core, we make use of unary encoding paradigm and orthogonal quantum layers, and introduce a new quantum Fourier transform in the unary basis. With respect to the number of samples, our quantum algorithm is proven to be substantially faster than the classical counterpart. We benchmark our proposed algorithms on three PDE families, namely Burger's equation, Darcy's flow equation and the Navier-Stokes equation, and the results show that our quantum methods are comparable in performance to the classical FNO. We also show an analysis of the image classification tasks where our proposed algorithms are able to match the accuracy of the CNNs, thereby showing their applicability to other domains.

1. INTRODUCTION

Solving Partial Differential Equations (PDEs) has been a very crucial step in understanding the dynamics of nature. They have been widely used to understand natural phenomenon such as heattransfer, modelling the flow of fluids, electromagnetism, etc. and lately have also found their applications in understanding the behavior of dynamical markets. In most of these applications, a closed form solution is difficult to find for the resulting PDEs and thus, classical solvers require a lot of evaluations to model the solution for a given PDE. To approximate such not-so-easily solvable PDE's, there has been an extensive research based on neural networks. A group of these methods Yu et al. ( 2018 2020) addressed both these issues and posed the problem as learning a function-to-function mapping for parametric PDEs. Experiments on widely popular PDEs showed that it was effective in learning the mapping from a parametric initial condition function to the solution operator for a family of PDEs. The method proposes a Fourier layer, which uses a learnable linear transform sandwiched between a Fourier Transform (F T ) and an Inverse Fourier Transform (IF T ) operation. This is similar to the convolution operation as it also translates to multiplication in the Fourier space. The major bottleneck which might hinder the scalability of this classical Fourier Neural Operator (FNO) is its time complexity, limited by the classical Fourier Transform (FT) and Inverse Fourier Transform (IFT) operations inside the Fourier Layer. As shown previously by Musk (2020), these operations are much faster when deployed on quantum hardware. Similar advantages have led to significant developments in learning approaches based on near term quantum computing. The initial demonstrations of these algorithms involved experiments on a small-scale hardware Farhi & Neven 1



); Raissi et al. (2019); Bar & Sochen (2019) aimed at learning the solution function for an instance of PDE and thus requires to be re-trained every time the parameters/conditions in the PDE change. The other set of methods Zhu & Zabaras (2018); Adler & Öktem (2017); Bhatnagar et al. (2019) targeted at learning over a family of PDEs but for a specific resolution dependent, making these methods limited to the discretization or the sampling density used in the training data. A recent work Li et al. (

