Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets

• URL: http://arxiv.org/abs/2010.14324v1
• Date: Tue, 27 Oct 2020 14:34:54 GMT
• Title: Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets
• Authors: Christian Soize and Roger Ghanem
• Abstract summary: A novel extension of the Probabilistic Learning on Manifolds (PLoM) is presented. It makes it possible to synthesize solutions to a wide range of nonlinear boundary value problems. Three applications are presented.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A novel extension of the Probabilistic Learning on Manifolds (PLoM) is presented. It makes it possible to synthesize solutions to a wide range of nonlinear stochastic boundary value problems described by partial differential equations (PDEs) for which a stochastic computational model (SCM) is available and depends on a vector-valued random control parameter. The cost of a single numerical evaluation of this SCM is assumed to be such that only a limited number of points can be computed for constructing the training dataset (small data). Each point of the training dataset is made up realizations from a vector-valued stochastic process (the stochastic solution) and the associated random control parameter on which it depends. The presented PLoM constrained by PDE allows for generating a large number of learned realizations of the stochastic process and its corresponding random control parameter. These learned realizations are generated so as to minimize the vector-valued random residual of the PDE in the mean-square sense. Appropriate novel methods are developed to solve this challenging problem. Three applications are presented. The first one is a simple uncertain nonlinear dynamical system with a nonstationary stochastic excitation. The second one concerns the 2D nonlinear unsteady Navier-Stokes equations for incompressible flows in which the Reynolds number is the random control parameter. The last one deals with the nonlinear dynamics of a 3D elastic structure with uncertainties. The results obtained make it possible to validate the PLoM constrained by stochastic PDE but also provide further validation of the PLoM without constraint.

Related papers

• Learning Controlled Stochastic Differential Equations [61.82896036131116]
  This work proposes a novel method for estimating both drift and diffusion coefficients of continuous, multidimensional, nonlinear controlled differential equations with non-uniform diffusion. We provide strong theoretical guarantees, including finite-sample bounds for (L2), (Linfty), and risk metrics, with learning rates adaptive to coefficients' regularity. Our method is available as an open-source Python library.
  arXiv  Detail & Related papers  (2024-11-04T11:09:58Z)
• Total Uncertainty Quantification in Inverse PDE Solutions Obtained with Reduced-Order Deep Learning Surrogate Models [50.90868087591973]
  We propose an approximate Bayesian method for quantifying the total uncertainty in inverse PDE solutions obtained with machine learning surrogate models. We test the proposed framework by comparing it with the iterative ensemble smoother and deep ensembling methods for a non-linear diffusion equation.
  arXiv  Detail & Related papers  (2024-08-20T19:06:02Z)
• Weak Collocation Regression for Inferring Stochastic Dynamics with L\'{e}vy Noise [8.15076267771005]
  We propose a weak form of the Fokker-Planck (FP) equation for extracting dynamics with L'evy noise. Our approach can simultaneously distinguish mixed noise types, even in multi-dimensional problems.
  arXiv  Detail & Related papers  (2024-03-13T06:54:38Z)
• High-dimensional scaling limits and fluctuations of online least-squares SGD with smooth covariance [16.652085114513273]
  We derive high-dimensional scaling limits and fluctuations for the online least-squares Gradient Descent (SGD) algorithm. Our results have several applications, including characterization of the limiting mean-square estimation or prediction errors and their fluctuations.
  arXiv  Detail & Related papers  (2023-04-03T03:50:00Z)
• Score-based Diffusion Models in Function Space [140.792362459734]
  Diffusion models have recently emerged as a powerful framework for generative modeling. We introduce a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space. We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
  arXiv  Detail & Related papers  (2023-02-14T23:50:53Z)
• Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation [59.45669299295436]
  We propose a Monte Carlo PDE solver for training unsupervised neural solvers. We use the PDEs' probabilistic representation, which regards macroscopic phenomena as ensembles of random particles. Our experiments on convection-diffusion, Allen-Cahn, and Navier-Stokes equations demonstrate significant improvements in accuracy and efficiency.
  arXiv  Detail & Related papers  (2023-02-10T08:05:19Z)
• Fully probabilistic deep models for forward and inverse problems in parametric PDEs [1.9599274203282304]
  We introduce a physics-driven deep latent variable model (PDDLVM) to learn simultaneously parameter-to-solution (forward) and solution-to- parameter (inverse) maps of PDEs. The proposed framework can be easily extended to seamlessly integrate observed data to solve inverse problems and to build generative models. We demonstrate the efficiency and robustness of our method on finite element discretized parametric PDE problems.
  arXiv  Detail & Related papers  (2022-08-09T15:40:53Z)
• Inverting brain grey matter models with likelihood-free inference: a tool for trustable cytoarchitecture measurements [62.997667081978825]
  characterisation of the brain grey matter cytoarchitecture with quantitative sensitivity to soma density and volume remains an unsolved challenge in dMRI. We propose a new forward model, specifically a new system of equations, requiring a few relatively sparse b-shells. We then apply modern tools from Bayesian analysis known as likelihood-free inference (LFI) to invert our proposed model.
  arXiv  Detail & Related papers  (2021-11-15T09:08:27Z)
• Learning stochastic dynamical systems with neural networks mimicking the Euler-Maruyama scheme [14.436723124352817]
  We propose a data driven approach where parameters of the SDE are represented by a neural network with a built-in SDE integration scheme. The algorithm is applied to the geometric brownian motion and a version of the Lorenz-63 model.
  arXiv  Detail & Related papers  (2021-05-18T11:41:34Z)
• Probabilistic Numeric Convolutional Neural Networks [80.42120128330411]
  Continuous input signals like images and time series that are irregularly sampled or have missing values are challenging for existing deep learning methods. We propose Probabilistic Convolutional Neural Networks which represent features as Gaussian processes (GPs) We then define a convolutional layer as the evolution of a PDE defined on this GP, followed by a nonlinearity. In experiments we show that our approach yields a $3times$ reduction of error from the previous state of the art on the SuperPixel-MNIST dataset and competitive performance on the medical time2012 dataset PhysioNet.
  arXiv  Detail & Related papers  (2020-10-21T10:08:21Z)
• Deep-learning of Parametric Partial Differential Equations from Sparse and Noisy Data [2.4431531175170362]
  In this work, a new framework, which combines neural network, genetic algorithm and adaptive methods, is put forward to address all of these challenges simultaneously. A trained neural network is utilized to calculate derivatives and generate a large amount of meta-data, which solves the problem of sparse noisy data. Next, genetic algorithm is utilized to discover the form of PDEs and corresponding coefficients with an incomplete candidate library. A two-step adaptive method is introduced to discover parametric PDEs with spatially- or temporally-varying coefficients.
  arXiv  Detail & Related papers  (2020-05-16T09:09:57Z)

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.