In this work we consider the regularization of a supervised learning problem by partial differential equations (PDEs) and derive error bounds for the obtained approximation in terms of a PDE error term and a data error term. Assuming that the target function satisfies an unknown PDE, the PDE error term quantifies how well this PDE is approximated by the auxiliary PDE used for regularization. It is shown that this error term decreases if more data is provided. The data error term quantifies the accuracy of the given data. Furthermore, the PDE-regularized learning problem is discretized by generalized Galerkin discretizations solving the associated minimization problem in subsets of the infinite dimensional functions space, which are not necessarily subspaces. For such discretizations an error bound in terms of the PDE error, the data error, and a best approximation error is derived.
We prove the support recovery for a general class of linear and nonlinear evolutionary partial differential equation (PDE) identification from a single noisy trajectory using $\ell_1$ regularized Pseudo-Least Squares model~($\ell_1$-PsLS). In any associative $\mathbb{R}$-algebra generated by finitely many differentiation operators that contain the unknown PDE operator, applying $\ell_1$-PsLS to a given data set yields a family of candidate models with coefficients $\mathbf{c}(λ)$ parameterized by the regularization weight $λ\geq 0$. The trace of $\{\mathbf{c}(λ)\}_{λ\geq 0}$ suffers from high …
A regularized truncated finite element method for degenerate parabolic stochastic PDE on non-compact graph
The Hong Kong Polytechnic University（香港理工大学）；Chalmers（查尔姆斯理工大学）；University of Technology and Economics（布达佩斯科技经济大学）；Pázmány Péter Catholic University（帕兹马尼·彼得天主教大学）；CAS AMSS-PolyU Joint Laboratory of Applied Mathematics（中科院应用数学与香港理工大学联合实验室）
The problem of solving partial differential equations (PDEs) can be formulated into a least-squares minimization problem, where neural networks are used to parametrize PDE solutions. A global minimizer corresponds to a neural network that solves the given PDE. In this paper, we show that the gradient descent method can identify a global minimizer of the least-squares optimization for solving second-order linear PDEs with two-layer neural networks under the assumption of over-parametrization. We also analyze the generalization error of the least-squares optimization for second-order linear PDEs…
This paper proves error estimates for $H^2$ conforming finite elements for equations which model the flow of surfaces by different powers of the mean curvature (this includes mean curvature flow). for an adapted scheme originally proposed in [17] for the inverse mean curvature flow. The scheme is based on a known regularization procedure and produces different kinds of errors, a regularization error, a finite element discretization error for the regularized problems and a full error. While in the literature and own previous work different aspects of the aforementioned error types are treated, …
Recent quasi-optimal error estimates for the finite element approximation of total-variation regularized minimization problems using the Crouzeix--Raviart finite element require the existence of a Lipschitz continuous dual solution, which is not generally given. We provide analytic proofs showing that the Lipschitz continuity of a dual solution is not necessary, in general. Using the Lipschitz truncation technique, we, in addition, derive error estimates that depend directly on the Sobolev regularity of a given dual solution.
We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of stochastic Galerkin finite element methods. We perform a posteriori error analysis of Galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations. Practical versions of this error estimate are discussed and tested numerically for a model problem with non-affine parametric representation of the coefficient. Furthermore, we use the error reduction indica…
The Alternating Direction Method of Multipliers (ADMM) provides a natural way of solving inverse problems with multiple partial differential equations (PDE) forward models and nonsmooth regularization. ADMM allows splitting these large-scale inverse problems into smaller, simpler sub-problems, for which computationally efficient solvers are available. In particular, we apply large-scale second-order optimization methods to solve the fully-decoupled Tikhonov regularized inverse problems stemming from each PDE forward model. We use fast proximal methods to handle the nonsmooth regularization ter…
This paper establishes the optimal $H^1$-norm error estimate for a nonstandard finite element method for approximating $H^2$ strong solutions of second order linear elliptic PDEs in non-divergence form with continuous coefficients. To circumvent the difficulty of lacking an effective duality argument for this class of PDEs, a new analysis technique is introduced; the crux of it is to establish an $H^1$-norm stability estimate for the finite element approximation operator which mimics a similar estimate for the underlying PDE operator recently established by the authors and its proof is based o…
Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work, we show a priori rates of convergence of this approximation process in standard Sobolev norms, with minimal regularity assumptions on the approximation of the Dirac delta distribution. The application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp $H^1$ and $L^2$ error estimates for the corresponding regularized problem. As an …
When time-dependent partial differential equations (PDEs) are solved numerically in a domain with curved boundary or on a curved surface, mesh error and geometric approximation error caused by the inaccurate location of vertices and other interior grid points, respectively, could be the main source of the inaccuracy and instability of the numerical solutions of PDEs. The role of these geometric errors in deteriorating the stability and particularly the conservation properties are largely unknown, which seems to necessitate very fine meshes especially to remove geometric approximation error. Th…
Multigoal-oriented optimal control problems with nonlinear PDE constraints
Johannes Kepler University（约翰·开普勒大学）；Austrian Academy of Sciences（奥地利科学院）；Leibniz Universität Hannover（汉诺威莱布尼茨大学）；Technische Universität Darmstadt（达姆施塔特工业大学）
We consider the discretization of elliptic boundary-value problems by variational physics-informed neural networks (VPINNs), in which test functions are continuous, piecewise linear functions on a triangulation of the domain. We define an a posteriori error estimator, made of a residual-type term, a loss-function term, and data oscillation terms. We prove that the estimator is both reliable and efficient in controlling the energy norm of the error between the exact and VPINN solutions. Numerical results are in excellent agreement with the theoretical predictions.
Machine learned partial differential equation (PDE) solvers trade the reliability of standard numerical methods for potential gains in accuracy and/or speed. The only way for a solver to guarantee that it outputs the exact solution is to use a convergent method in the limit that the grid spacing $Δx$ and timestep $Δt$ approach zero. Machine learned solvers, which learn to update the solution at large $Δx$ and/or $Δt$, can never guarantee perfect accuracy. Some amount of error is inevitable, so the question becomes: how do we constrain machine learned solvers to give us the sorts of errors that…
In our previous work [Singler, SIAM J. Numer. Anal. 52 (2014), no. 2, 852-876], we considered the proper orthogonal decomposition (POD) of time varying PDE solution data taking values in two different Hilbert spaces. We considered various POD projections of the data and obtained new results concerning POD projection errors and error bounds for POD reduced order models of PDEs. In this work, we improve on our earlier results concerning POD projections by extending to a more general framework that allows for non-orthogonal POD projections and seminorms. We obtain new exact error formulas and con…
Error estimates of a regularized finite difference method for the logarithmic Schrödinger equation
National University of Singapore（新加坡国立大学）；CNRS, Institut Montpelliérain Alexander Grothendieck, Univ. Montpellier（法国国家科学研究中心，蒙彼利埃亚历山大·格罗滕迪克研究所，蒙彼利埃大学）；University of Innsbruck（因斯布鲁克大学）；Sichuan University（四川大学）
In this work, we investigate the regularized solutions and their finite element solutions to the inverse source problems governed by partial differential equations, and establish the stochastic convergence and optimal finite element convergence rates of these solutions, under pointwise measurement data with random noise. Unlike most existing regularization theories, the regularization error estimates are derived without any source conditions, while the error estimates of finite element solutions show their explicit dependence on the noise level, regularization parameter, mesh size, and time st…
We consider a diffuse interface approach for solving an elliptic PDE on a given closed hypersurface. The method is based on a (bulk) finite element scheme employing numerical quadrature for the phase field function and hence is very easy to implement compared to other approaches. We estimate the error in natural norms in terms of the spatial grid size, the interface width and the order of the underlying quadrature rule. Numerical test calculations are presented which confirm the form of the error bounds.
In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition, by establishing the upper bounds on the number of training samples, depth and width of the deep neural networks to achieve desired accuracy. The error of PINNs is decomposed into approximation error and statistical error, where the approximation error is given in $C^2$ norm with $\mathrm{…
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 fo…
Nonlinear model predictive control (NMPC) often requires real-time solution to optimization problems. However, in cases where the mathematical model is of high dimension in the solution space, e.g. for solution of partial differential equations (PDEs), black-box optimizers are rarely sufficient to get the required online computational speed. In such cases one must resort to customized solvers. This paper present a new solver for nonlinear time-dependent PDE-constrained optimization problems. It is composed of a sequential quadratic programming (SQP) scheme to solve the PDE-constrained problem …
Sharp Error Bounds for Piecewise Polynomial Approximation: Revisit and Application to Elliptic PDE Eigenvalue Computation
LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所）；Beijing Computational Science Research Center（北京计算科学研究中心）；Department of Mathematics, Wayne State University（韦恩州立大学数学系）
We introduce an unfitted finite element method with Lagrange-multipliers to study an Eulerian time stepping scheme for moving domain problems applied to a model problem where the domain motion is implicit to the problem. We consider a parabolic partial differential equation (PDE) in the bulk domain, and the domain motion is described by an ordinary differential equation (ODE), coupled to the bulk partial differential equation through the transfer of forces at the moving interface. The discretisation is based on an unfitted finite element discretisation on a time-independent mesh. The method-of…
Error Bounds for Physics-Informed Neural Networks in Fokker-Planck PDEs
Dept. of Aerospace Eng. Sciences University of Colorado Boulder（航空航天工程科学系，科罗拉多大学博尔德分校）；Center for Systems and Control Delft University of Technology（系统与控制中心，代尔夫特理工大学）
