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Register a new userFriedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning
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Haizhao Yang
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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This paper proposes Friedrichs learning as a novel deep learning methodology that can learn the weak solutions of PDEs via a minmax formulation, which transforms the PDE problem into a minimax optimization problem to identify weak solutions. The name Friedrichs learning is for highlighting the close relationship between our learning strategy and Friedrichs theory on symmetric systems of PDEs. The weak solution and the test function in the weak formulation are parameterized as deep neural networks in a mesh-free manner, which are alternately updated to approach the optimal solution networks approximating the weak solution and the optimal test function, respectively. Extensive numerical results indicate that our mesh-free method can provide reasonably good solutions to a wide range of PDEs defined on regular and irregular domains in various dimensions, where classical numerical methods such as finite difference methods and finite element methods may be tedious or difficult to be applied.
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Sparse Identification of Nonlinear Dynamics (SINDy) is a method of system discovery that has been shown to successfully recover governing dynamical systems from data (Brunton et al., PNAS, 16; Rudy et al., Sci. Adv. 17). Recently, several groups have independently discovered that the weak formulation provides orders of magnitude better robustness to noise. Here we extend our Weak SINDy (WSINDy) framework introduced in (arXiv:2005.04339) to the setting of partial differential equations (PDEs). The elimination of pointwise derivative approximations via the weak form enables effective machine-precision recovery of model coefficients from noise-free data (i.e. below the tolerance of the simulation scheme) as well as robust identification of PDEs in the large noise regime (with signal-to-noise ratio approaching one in many well-known cases). This is accomplished by discretizing a convolutional weak form of the PDE and exploiting separability of test functions for efficient model identification using the Fast Fourier Transform. The resulting WSINDy algorithm for PDEs has a worst-case computational complexity of $mathcal{O}(N^{D+1}log(N))$ for datasets with $N$ points in each of $D+1$ dimensions (i.e. $mathcal{O}(log(N))$ operations per datapoint). Furthermore, our Fourier-based implementation reveals a connection between robustness to noise and the spectra of test functions, which we utilize in an textit{a priori} selection algorithm for test functions. Finally, we introduce a learning algorithm for the threshold in sequential-thresholding least-squares (STLS) that enables model identification from large libraries, and we utilize scale-invariance at the continuum level to identify PDEs from poorly-scaled datasets. We demonstrate WSINDys robustness, speed and accuracy on several challenging PDEs.
We present a neural network-based method for solving linear and nonlinear partial differential equations, by combining the ideas of extreme learning machines (ELM), domain decomposition and local neural networks. The field solution on each sub-domain is represented by a local feed-forward neural network, and $C^k$ continuity is imposed on the sub-domain boundaries. Each local neural network consists of a small number of hidden layers, while its last hidden layer can be wide. The weight/bias coefficients in all hidden layers of the local neural networks are pre-set to random values and are fixed, and only the weight coefficients in the output layers are training parameters. The overall neural network is trained by a linear or nonlinear least squares computation, not by the back-propagation type algorithms. We introduce a block time-marching scheme together with the presented method for long-time dynamic simulations. The current method exhibits a clear sense of convergence with respect to the degrees of freedom in the neural network. Its numerical errors typically decrease exponentially or nearly exponentially as the number of degrees of freedom increases. Extensive numerical experiments have been performed to demonstrate the computational performance of the presented method. We compare the current method with the deep Galerkin method (DGM) and the physics-informed neural network (PINN) in terms of the accuracy and computational cost. The current method exhibits a clear superiority, with its numerical errors and network training time considerably smaller (typically by orders of magnitude) than those of DGM and PINN. We also compare the current method with the classical finite element method (FEM). The computational performance of the current method is on par with, and oftentimes exceeds, the FEM performance.
We propose an accurate data-driven numerical scheme to solve Stochastic Differential Equations (SDEs), by taking large time steps. The SDE discretization is built up by means of a polynomial chaos expansion method, on the basis of accurately determined stochastic collocation (SC) points. By employing an artificial neural network to learn these SC points, we can perform Monte Carlo simulations with large time steps. Error analysis confirms that this data-driven scheme results in accurate SDE solutions in the sense of strong convergence, provided the learning methodology is robust and accurate. With a variant method called the compression-decompression collocation and interpolation technique, we can drastically reduce the number of neural network functions that have to be learned, so that computational speed is enhanced. Numerical results show the high quality strong convergence error results, when using large time steps, and the novel scheme outperforms some classical numerical SDE discretizations. Some applications, here in financial option valuation, are also presented.
Random ordinary differential equations (RODEs), i.e. ODEs with random parameters, are often used to model complex dynamics. Most existing methods to identify unknown governing RODEs from observed data often rely on strong prior knowledge. Extracting the governing equations from data with less prior knowledge remains a great challenge. In this paper, we propose a deep neural network, called RODE-Net, to tackle such challenge by fitting a symbolic expression of the differential equation and the distribution of parameters simultaneously. To train the RODE-Net, we first estimate the parameters of the unknown RODE using the symbolic networks cite{long2019pde} by solving a set of deterministic inverse problems based on the measured data, and use a generative adversarial network (GAN) to estimate the true distribution of the RODEs parameters. Then, we use the trained GAN as a regularization to further improve the estimation of the ODEs parameters. The two steps are operated alternatively. Numerical results show that the proposed RODE-Net can well estimate the distribution of model parameters using simulated data and can make reliable predictions. It is worth noting that, GAN serves as a data driven regularization in RODE-Net and is more effective than the $ell_1$ based regularization that is often used in system identifications.
Fast and accurate solutions of time-dependent partial differential equations (PDEs) are of pivotal interest to many research fields, including physics, engineering, and biology. Generally, implicit/semi-implicit schemes are preferred over explicit ones to improve stability and correctness. However, existing semi-implicit methods are usually iterative and employ a general-purpose solver, which may be sub-optimal for a specific class of PDEs. In this paper, we propose a neural solver to learn an optimal iterative scheme in a data-driven fashion for any class of PDEs. Specifically, we modify a single iteration of a semi-implicit solver using a deep neural network. We provide theoretical guarantees for the correctness and convergence of neural solvers analogous to conventional iterative solvers. In addition to the commonly used Dirichlet boundary condition, we adopt a diffuse domain approach to incorporate a diverse type of boundary conditions, e.g., Neumann. We show that the proposed neural solver can go beyond linear PDEs and applies to a class of non-linear PDEs, where the non-linear component is non-stiff. We demonstrate the efficacy of our method on 2D and 3D scenarios. To this end, we show how our model generalizes to parameter settings, which are different from training; and achieves faster convergence than semi-implicit schemes.
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