From Monte Carlo to neural networks approximations of boundary value problems

• URL: http://arxiv.org/abs/2209.01432v3
• Date: Sat, 10 Aug 2024 04:55:50 GMT
• Title: From Monte Carlo to neural networks approximations of boundary value problems
• Authors: Lucian Beznea, Iulian Cimpean, Oana Lupascu-Stamate, Ionel Popescu, Arghir Zarnescu,
• Abstract summary: We show that the solution to Poisson equation can be numerically approximated in the sup-norm by Monte Carlo methods. We also show that the obtained Monte Carlo solver renders in a constructive way ReLU deep neural network (DNN) solutions to Poisson problem.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper we study probabilistic and neural network approximations for solutions to Poisson equation subject to Holder data in general bounded domains of $\mathbb{R}^d$. We aim at two fundamental goals. The first, and the most important, we show that the solution to Poisson equation can be numerically approximated in the sup-norm by Monte Carlo methods, and that this can be done highly efficiently if we use a modified version of the walk on spheres algorithm as an acceleration method. This provides estimates which are efficient with respect to the prescribed approximation error and with polynomial complexity in the dimension and the reciprocal of the error. A crucial feature is that the overall number of samples does not not depend on the point at which the approximation is performed. As a second goal, we show that the obtained Monte Carlo solver renders in a constructive way ReLU deep neural network (DNN) solutions to Poisson problem, whose sizes depend at most polynomialy in the dimension $d$ and in the desired error. In fact we show that the random DNN provides with high probability a small approximation error and low polynomial complexity in the dimension.

Related papers

• Solving Poisson Equations using Neural Walk-on-Spheres [80.1675792181381]
  We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. We demonstrate the superiority of NWoS in accuracy, speed, and computational costs.
  arXiv  Detail & Related papers  (2024-06-05T17:59:22Z)
• A Mean-Field Analysis of Neural Stochastic Gradient Descent-Ascent for Functional Minimax Optimization [90.87444114491116]
  This paper studies minimax optimization problems defined over infinite-dimensional function classes of overparametricized two-layer neural networks. We address (i) the convergence of the gradient descent-ascent algorithm and (ii) the representation learning of the neural networks. Results show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of $O(alpha-1)$, measured in terms of the Wasserstein distance.
  arXiv  Detail & Related papers  (2024-04-18T16:46:08Z)
• Polynomial-Time Solutions for ReLU Network Training: A Complexity Classification via Max-Cut and Zonotopes [70.52097560486683]
  We prove that the hardness of approximation of ReLU networks not only mirrors the complexity of the Max-Cut problem but also, in certain special cases, exactly corresponds to it. In particular, when $epsilonleqsqrt84/83-1approx 0.006$, we show that it is NP-hard to find an approximate global dataset of the ReLU network objective with relative error $epsilon$ with respect to the objective value.
  arXiv  Detail & Related papers  (2023-11-18T04:41:07Z)
• A multiobjective continuation method to compute the regularization path of deep neural networks [1.3654846342364308]
  Sparsity is a highly feature in deep neural networks (DNNs) since it ensures numerical efficiency, improves the interpretability of models, and robustness. We present an algorithm that allows for the entire sparse front for the above-mentioned objectives in a very efficient manner for high-dimensional gradients with millions of parameters. We demonstrate that knowledge of the regularization path allows for a well-generalizing network parametrization.
  arXiv  Detail & Related papers  (2023-08-23T10:08:52Z)
• Bayesian polynomial neural networks and polynomial neural ordinary differential equations [4.550705124365277]
  Symbolic regression with neural networks and neural ordinary differential equations (ODEs) are powerful approaches for equation recovery of many science and engineering problems. These methods provide point estimates for the model parameters and are currently unable to accommodate noisy data. We address this challenge by developing and validating the following inference methods: the Laplace approximation, Markov Chain Monte Carlo sampling methods, and Bayesian variational inference.
  arXiv  Detail & Related papers  (2023-08-17T05:42:29Z)
• Advancing Algorithm to Scale and Accurately Solve Quantum Poisson Equation on Near-term Quantum Hardware [0.0]
  We present an advanced quantum algorithm for solving the Poisson equation with high accuracy and dynamically tunable problem size. Particularly, in this work we present an advanced circuit that ensures the accuracy of the solution by implementing non-truncated eigenvalues. We show that our algorithm not only increases the accuracy of the solutions but also composes more practical and scalable circuits.
  arXiv  Detail & Related papers  (2022-10-29T18:50:40Z)
• Efficient semidefinite-programming-based inference for binary and multi-class MRFs [83.09715052229782]
  We propose an efficient method for computing the partition function or MAP estimate in a pairwise MRF. We extend semidefinite relaxations from the typical binary MRF to the full multi-class setting, and develop a compact semidefinite relaxation that can again be solved efficiently using the solver.
  arXiv  Detail & Related papers  (2020-12-04T15:36:29Z)
• Online Model Selection for Reinforcement Learning with Function Approximation [50.008542459050155]
  We present a meta-algorithm that adapts to the optimal complexity with $tildeO(L5/6 T2/3)$ regret. We also show that the meta-algorithm automatically admits significantly improved instance-dependent regret bounds.
  arXiv  Detail & Related papers  (2020-11-19T10:00:54Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)
• Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach [14.558626910178127]
  The eigenvalue problem is reformulated as a fixed point problem of the semigroup flow induced by the operator. The method shares a similar spirit with diffusion Monte Carlo but augments a direct approximation to the eigenfunction through neural-network ansatz. Our approach is able to provide accurate eigenvalue and eigenfunction approximations in several numerical examples.
  arXiv  Detail & Related papers  (2020-02-07T03:08:31Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.