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 We also use a similar idea to do the multi-channel intermediate linear transform using orthogonal matrix but using parameterized butterfly circuits instead of pyramid circuits. Exploiting these advantages offered by quantum computing, in this work, we propose a new kind of Quantum Fourier Transform (QFT), which operates on the unary states and a learnable Quantum Linear Transform. We further propose three quantum algorithms inspired by this classical Fourier operation which are faster than the classical operation and require fewer parameters for the same architecture, thereby boosting their scalability. Given the input of dimension N s × N c , where N s corresponds to number of samples per PDE and N c correspond to feature dimension, the order of time complexity corresponding to Fourier Layer (FL) and proposed algorithms is shown in table 1 . Table 1: Comparison of order of time/depth complexities (O) of the proposed circuits with the existing classical Fourier Layer (FL). Here Ns denote the sampling dimension, Nc denote the feature dimension where Ns Nc and K (usually in range 4-16) denotes the maximum number of modes allowed Li et al. (2020) . This implies that the proposed quantum algorithms would be faster than the classical method. Each quantum circuit requires Nc + Ns qubits and K independent parallel circuits are required by the Parallelized QFNO.

Method

Classical FL Sequential Quantum FL Parallel Quantum FL Compound Quantum FL Complexity N c +N s log(N s ) Klog(N c )+N c log(N s ) log(N c )+N c log(N s ) log(N c +K)+N c log(N s ) # qubits - N c + N s N c + N s N c + N s # parallel circuits - 1 K 1 The first algorithm replicates the classical operation on a quantum circuit. The other two algorithms are modifications of the first circuit designed for the noisy learning process offered by the near term quantum hardware. We test all the three proposed algorithms on all the three PDEs evaluated To formulate it, given two functional spaces A and U along with a set of observations {a j , u j } (a j ∼ µ is an i.i.d. sequence sampled from some function f ∈ A), it learns a parametric mapping G : A × Θ → U. To achieve this, it proposed a learning network based on iteratively applying a new kind of layer which it termed as the Fourier Layer. The layer consists of two parts, the top part involves firstly projecting the input to the Fourier domain and then applying



); 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. (

2018); Coyle et al. (2020); Cappelletti et al. (2020); Grant et al. (2018) which established their effectiveness in extracting patterns. Following this, many works Abbas et al. (2021); Mari et al. (2020); Beer et al. (2020); Allcock et al. (2020) proposed small-scale implementations of fully connected quantum neural networks on near term hardware. Other proposals Kerenidis et al. (2019); Cong et al. (2019) for deploying convolution-based learning methods on quantum devices showed effective training in practice. Furthermore, Chakrabarti et al. (2019) proposed quantum-hardware implementation for generative adversarial networks. A different approach, where the inputs are encoded as unary states, using the two-qubits quantum gate RBS (Reconfigurable Beam Splitter) was proposed in a recent work Johri et al. (2021). This encoding gave rise to the use of orthogonal properties of pure quantum unitaries, as proposed in Kerenidis et al. (2021) for training, for instance, orthogonal feed-forward networks to damp the gradient based issues while learning. It used a pyramid circuit based on parameterized RBS gates to implement a learnable orthogonal matrix as compared to the existing classical approaches which offer approximate orthogonality at the cost of increased training time. This orthogonality in neural networks results in much smoother convergence and also lesser parameters as shown by Li et al. (2019) for feed-forward neural networks and Wang et al. (2020) for convolutional nets. The effectiveness of these orthogonal quantum networks was further shown in another work on medical image classification Mathur et al. (2021) problem.

in the classical FNO paper Li et al. (2020) namely the Burgers equation, Darcy's Flow equation and Navier Stokes equation on the synthetic datasets used in that paper. We also test our algorithms against the Convolutional neural networks (CNNs) on benchmark datasets for image classification namely MNIST, Fashion-MNIST Xiao et al. (2017), Pneumonia-MNIST Yang et al. (2021). In all the experiments, three algorithms perform similarly and comparable to state-of-the-art FNO for PDEs. Also, they perform decently on the image classification tasks. 2 CLASSICAL FOURIER NEURAL OPERATOR Given a training set comprising the family of a Partial Differential Equation, the classical FNO Li et al. (2020) aims to learn a functional mapping from a parameterized initial condition to the solution function for this family. This means given an initial condition function characterizing a PDE instance, sampled at different points, it should predict the solution function values at those points at the inference time.

