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Register a new userOptimally weighted loss functions for solving PDEs with Neural Networks
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Added by
Anastasia Borovykh
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a scaling parameter which balances the relative importance of the different constraints imposed by partial differential equations. A mathematical motivation of these generalized methods is provided, which shows that for linear and well-posed partial differential equations, the functional form is convex. We then derive a choice for the scaling parameter that is optimal with respect to a measure of relative error. Because this optimal choice relies on having full knowledge of analytical solutions, we also propose a heuristic method to approximate this optimal choice. The proposed methods are compared numerically to the original methods on a variety of model partial differential equations, with the number of data points being updated adaptively. For several problems, including high-dimensional PDEs the proposed methods are shown to significantly enhance accuracy.
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We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian processes. The proposed approach: (1) provides a natural generalization of collocation kernel methods to nonlinear PDEs and IPs; (2) has guaranteed convergence for a very general class of PDEs, and comes equipped with a path to compute error bounds for specific PDE approximations; (3) inherits the state-of-the-art computational complexity of linear solvers for dense kernel matrices. The main idea of our method is to approximate the solution of a given PDE as the maximum a posteriori (MAP) estimator of a Gaussian process conditioned on solving the PDE at a finite number of collocation points. Although this optimization problem is infinite-dimensional, it can be reduced to a finite-dimensional one by introducing additional variables corresponding to the values of the derivatives of the solution at collocation points; this generalizes the representer theorem arising in Gaussian process regression. The reduced optimization problem has the form of a quadratic objective function subject to nonlinear constraints; it is solved with a variant of the Gauss--Newton method. The resulting algorithm (a) can be interpreted as solving successive linearizations of the nonlinear PDE, and (b) in practice is found to converge in a small number of iterations (2 to 10), for a wide range of PDEs. Most traditional approaches to IPs interleave parameter updates with numerical solution of the PDE; our algorithm solves for both parameter and PDE solution simultaneously. Experiments on nonlinear elliptic PDEs, Burgers equation, a regularized Eikonal equation, and an IP for permeability identification in Darcy flow illustrate the efficacy and scope of our framework.
Data assisted reconstruction algorithms, incorporating trained neural networks, are a novel paradigm for solving inverse problems. One approach is to first apply a classical reconstruction method and then apply a neural network to improve its solution. Empirical evidence shows that such two-step methods provide high-quality reconstructions, but they lack a convergence analysis. In this paper we formalize the use of such two-step approaches with classical regularization theory. We propose data-consistent neural networks that we combine with classical regularization methods. This yields a data-driven regularization method for which we provide a full convergence analysis with respect to noise. Numerical simulations show that compared to standard two-step deep learning methods, our approach provides better stability with respect to structural changes in the test set, while performing similarly on test data similar to the training set. Our method provides a stable solution of inverse problems that exploits both the known nonlinear forward model as well as the desired solution manifold from data.
Recently, neural networks have been widely applied for solving partial differential equations. However, the resulting optimization problem brings many challenges for current training algorithms. This manifests itself in the fact that the convergence order that has been proven theoretically cannot be obtained numerically. In this paper, we develop a novel greedy training algorithm for solving PDEs which builds the neural network architecture adaptively. It is the first training algorithm that observes the convergence order of neural networks numerically. This innovative algorithm is tested on several benchmark examples in both 1D and 2D to confirm its efficiency and robustness.
In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM), and its local kernel variants, to approximate second-order differential operators defined on smooth manifolds with boundaries that naturally arise in elliptic PDE models. To achieve this goal, we introduce the Ghost Point Diffusion Maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points, specified along the estimated tangential direction at the sampled points at the boundary. The resulting GPDM estimator restricts the standard DM matrix to a set of extrapolation equations that estimates the function values at the ghost points. This adjustment is analogous to the classical ghost point method in finite-difference scheme for solving PDEs on flat domain. As opposed to the classical DM which diverges near the boundary, the proposed GPDM estimator converges pointwise even near the boundary. Applying the consistent GPDM estimator to solve the well-posed elliptic PDEs with classical boundary conditions (Dirichlet, Neumann, and Robin), we establish the convergence of the approximate solution under appropriate smoothness assumptions. We numerically validate the proposed mesh-free PDE solver on various problems defined on simple sub-manifolds embedded in Euclidean spaces as well as on an unknown manifold. Numerically, we also found that the GPDM is more accurate compared to DM in solving elliptic eigenvalue problems on bounded smooth manifolds.
Solving partial differential equations (PDEs) by parametrizing its solution by neural networks (NNs) has been popular in the past a few years. However, different types of loss functions can be proposed for the same PDE. For the Poisson equation, the loss function can be based on the weak formulation of energy variation or the least squares method, which leads to the deep Ritz model and deep Galerkin model, respectively. But loss landscapes from these different models give arise to different practical performance of training the NN parameters. To investigate and understand such practical differences, we propose to compare the loss landscapes of these models, which are both high dimensional and highly non-convex. In such settings, the roughness is more important than the traditional eigenvalue analysis to describe the non-convexity. We contribute to the landscape comparisons by proposing a roughness index to scientifically and quantitatively describe the heuristic concept of roughness of landscape around minimizers. This index is based on random projections and the variance of (normalized) total variation for one dimensional projected functions, and it is efficient to compute. A large roughness index hints an oscillatory landscape profile as a severe challenge for the first order optimization method. We apply this index to the two models for the Poisson equation and our empirical results reveal a consistent general observation that the landscapes from the deep Galerkin method around its local minimizers are less rough than the deep Ritz method, which supports the observed gain in accuracy of the deep Galerkin method.
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