It is one of the most challenging issues in applied mathematics to approximately solve high-dimensional partial differential equations (PDEs) and most of the numerical approximation methods for PDEs in the scientific literature suffer from the so-called curse of dimensionality in the sense that the number of computational operations employed in the corresponding approximation scheme to obtain an approximation precision $\varepsilon>0$ grows exponentially in the PDE dimension and/or the reciprocal of $\varepsilon$. Recently, certain deep learning based approximation methods for PDEs have been proposed and various numerical simulations for such methods suggest that deep neural network (DNN) approximations might have the capacity to indeed overcome the curse of dimensionality in the sense that the number of real parameters used to describe the approximating DNNs grows at most polynomially in both the PDE dimension $d\in\mathbb{N}$ and the reciprocal of the prescribed accuracy $\varepsilon>0$. There are now also a few rigorous results in the scientific literature which substantiate this conjecture by proving that DNNs overcome the curse of dimensionality in approximating solutions of PDEs. Each of these results establishes that DNNs overcome the curse of dimensionality in approximating suitable PDE solutions at a fixed time point $T>0$ and on a compact cube $[a,b]^d$ in space but none of these results provides an answer to the question whether the entire PDE solution on $[0,T]\times [a,b]^d$ can be approximated by DNNs without the curse of dimensionality. It is precisely the subject of this article to overcome this issue. More specifically, the main result of this work in particular proves for every $a\in\mathbb{R}$, $ b\in (a,\infty)$ that solutions of certain Kolmogorov PDEs can be approximated by DNNs on the space-time region $[0,T]\times [a,b]^d$ without the curse of dimensionality.
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We provide a polynomial-time classical algorithm for noisy quantum circuits. The algorithm computes the expectation value of any observable for any circuit, with a small average error over input states drawn from an ensemble (e.g. the computational basis). Our approach is based upon the intuition that noise exponentially damps non-local correlations relative to local correlations. This enables one to classically simulate a noisy quantum circuit by only keeping track of the dynamics of local quantum information. Our algorithm also enables sampling from the output distribution of a circuit in quasi-polynomial time, so long as the distribution anti-concentrates. A number of practical implications are discussed, including a fundamental limit on the efficacy of noise mitigation strategies: any quantum circuit for which error mitigation is efficient must be classically simulable.
We present a simple algorithm to approximate the viscosity solution of Hamilton-Jacobi (HJ) equations by means of an artificial deep neural network. The algorithm uses a stochastic gradient descent-based method to minimize the least square principle defined by a monotone, consistent numerical scheme. We analyze the least square principle's critical points and derive conditions that guarantee that any critical point approximates the sought viscosity solution. The use of a deep artificial neural network on a finite difference scheme lifts the restriction of conventional finite difference methods that rely on computing functions on a fixed grid. This feature makes it possible to solve HJ equations posed in higher dimensions where conventional methods are infeasible. We demonstrate the efficacy of our algorithm through numerical studies on various canonical HJ equations across different dimensions, showcasing its potential and versatility.
The Discrete Ordinates Method (DOM) is the most widely used velocity discretization method for simulating the radiative transport equation. The ray effect stands as a long-standing drawback of DOM. In benchmark tests displaying the ray effect, we observe low regularity in velocity within the solution. To address this issue, we propose a random ordinate method (ROM) to mitigate the ray effect. Compared with other strategies proposed in the literature for mitigating the ray effect, ROM offers several advantages: 1) the computational cost is comparable to DOM; 2) it is simple and requires minimal changes to existing code based on DOM; 3) it is easily parallelizable and independent of the problem setup. Analytical results are presented for the convergence orders of the error and bias, and numerical tests demonstrate its effectiveness in mitigating the ray effect.
We present a novel and mathematically transparent approach to function approximation and the training of large, high-dimensional neural networks, based on the approximate least-squares solution of associated Fredholm integral equations of the first kind by Ritz-Galerkin discretization, Tikhonov regularization and tensor-train methods. Practical application to supervised learning problems of regression and classification type confirm that the resulting algorithms are competitive with state-of-the-art neural network-based methods.
The stochastic description of chemical reaction networks with the kinetic chemical master equation (CME) is important for studying biological cells, but it suffers from the curse of dimensionality: The amount of data to be stored grows exponentially with the number of chemical species and thus exceeds the capacity of common computational devices for realistic problems. Therefore, time-dependent model order reduction techniques such as the dynamical low-rank approximation are desirable. In this paper we propose a dynamical low-rank algorithm for the kinetic CME using binary tree tensor networks. The dimensionality of the problem is reduced in this approach by hierarchically dividing the reaction network into partitions. Only reactions that cross partitions are subject to an approximation error. We demonstrate by two numerical examples (a 5-dimensional lambda phage model and a 20-dimensional reaction cascade) that the proposed method drastically reduces memory consumption and shows improved computational performance and better accuracy compared to a Monte Carlo method.
We propose a local discontinuous Galerkin (LDG) method for fractional Korteweg-de Vries equation involving the fractional Laplacian with exponent $\alpha\in (1,2)$ in one and two space dimensions. By decomposing the fractional Laplacian into a first order derivative and a fractional integral, we prove $L^2$-stability of the semi-discrete LDG scheme incorporating suitable interface and boundary fluxes. We analyze the error estimate by considering linear convection term and utilizing the estimate, we derive the error estimate for general nonlinear flux and demonstrate an order of convergence $\mathcal{O}(h^{k+1/2})$. Moreover, the stability and error analysis have been extended to multiple space dimensional case. Numerical illustrations are shown to demonstrate the efficiency of the scheme by obtaining an optimal order of convergence.
Predictive estimation, which comprises model calibration, model prediction, and validation, is a common objective when performing inverse uncertainty quantification (UQ) in diverse scientific applications. These techniques typically require thousands to millions of realisations of the forward model, leading to high computational costs. Surrogate models are often used to approximate these simulations. However, many surrogate models suffer from the fundamental limitation of being unable to estimate plausible high-dimensional outputs, inevitably compromising their use in the UQ framework. To address this challenge, this study introduces an efficient surrogate modelling workflow tailored for high-dimensional outputs. Specifically, a two-step approach is developed: (1) a dimensionality reduction technique is used for extracting data features and mapping the original output space into a reduced space; and (2) a multivariate surrogate model is constructed directly on the reduced space. The combined approach is shown to improve the accuracy of the surrogate model while retaining the computational efficiency required for UQ inversion. The proposed surrogate method, combined with Bayesian inference, is evaluated for a civil engineering application by performing inverse analyses on a laterally loaded pile problem. The results demonstrate the superiority of the proposed framework over traditional surrogate methods in dealing with high-dimensional outputs for sequential inversion analysis.
A global approximation method of Nystr\"om type is explored for the numerical solution of a class of nonlinear integral equations of the second kind. The cases of smooth and weakly singular kernels are both considered. In the first occurrence, the method uses a Gauss-Legendre rule whereas in the second one resorts to a product rule based on Legendre nodes. Stability and convergence are proved in functional spaces equipped with the uniform norm and several numerical tests are given to show the good performance of the proposed method. An application to the interior Neumann problem for the Laplace equation with nonlinear boundary conditions is also considered.
To solve many problems on graphs, graph traversals are used, the usual variants of which are the depth-first search and the breadth-first search. Implementing a graph traversal we consequently reach all vertices of the graph that belong to a connected component. The breadth-first search is the usual choice when constructing efficient algorithms for finding connected components of a graph. Methods of simple iteration for solving systems of linear equations with modified graph adjacency matrices and with the properly specified right-hand side can be considered as graph traversal algorithms. These traversal algorithms, generally speaking, turn out to be non-equivalent neither to the depth-first search nor the breadth-first search. The example of such a traversal algorithm is the one associated with the Gauss-Seidel method. For an arbitrary connected graph, to visit all its vertices, the algorithm requires not more iterations than that is required for BFS. For a large number of instances of the problem, fewer iterations will be required.
Neural ordinary differential equations (Neural ODEs) is a class of machine learning models that approximate the time derivative of hidden states using a neural network. They are powerful tools for modeling continuous-time dynamical systems, enabling the analysis and prediction of complex temporal behaviors. However, how to improve the model's stability and physical interpretability remains a challenge. This paper introduces new conservation relations in Neural ODEs using Lie symmetries in both the hidden state dynamics and the back propagation dynamics. These conservation laws are then incorporated into the loss function as additional regularization terms, potentially enhancing the physical interpretability and generalizability of the model. To illustrate this method, the paper derives Lie symmetries and conservation laws in a simple Neural ODE designed to monitor charged particles in a sinusoidal electric field. New loss functions are constructed from these conservation relations, demonstrating the applicability symmetry-regularized Neural ODE in typical modeling tasks, such as data-driven discovery of dynamical systems.
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