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Register a new userSolving Differential Equations Using Neural Network Solution Bundles
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The time evolution of dynamical systems is frequently described by ordinary differential equations (ODEs), which must be solved for given initial conditions. Most standard approaches numerically integrate ODEs producing a single solution whose values are computed at discrete times. When many varied solutions with different initial conditions to the ODE are required, the computational cost can become significant. We propose that a neural network be used as a solution bundle, a collection of solutions to an ODE for various initial states and system parameters. The neural network solution bundle is trained with an unsupervised loss that does not require any prior knowledge of the sought solutions, and the resulting object is differentiable in initial conditions and system parameters. The solution bundle exhibits fast, parallelizable evaluation of the system state, facilitating the use of Bayesian inference for parameter estimation in real dynamical systems.
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Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. The PINN algorithm is simple, and it can be applied to different types of PDEs, including integro-differential equations, fractional PDEs, and stochastic PDEs. Moreover, from the implementation point of view, PINNs solve inverse problems as easily as forward problems. We propose a new residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs. For pedagogical reasons, we compare the PINN algorithm to a standard finite element method. We also present a Python library for PINNs, DeepXDE, which is designed to serve both as an education tool to be used in the classroom as well as a research tool for solving problems in computational science and engineering. Specifically, DeepXDE can solve forward problems given initial and boundary conditions, as well as inverse problems given some extra measurements. DeepXDE supports complex-geometry domains based on the technique of constructive solid geometry, and enables the user code to be compact, resembling closely the mathematical formulation. We introduce the usage of DeepXDE and its customizability, and we also demonstrate the capability of PINNs and the user-friendliness of DeepXDE for five different examples. More broadly, DeepXDE contributes to the more rapid development of the emerging Scientific Machine Learning field.
We describe a neural-based method for generating exact or approximate solutions to differential equations in the form of mathematical expressions. Unlike other neural methods, our system returns symbolic expressions that can be interpreted directly. Our method uses a neural architecture for learning mathematical expressions to optimize a customizable objective, and is scalable, compact, and easily adaptable for a variety of tasks and configurations. The system has been shown to effectively find exact or approximate symbolic solutions to various differential equations with applications in natural sciences. In this work, we highlight how our method applies to partial differential equations over multiple variables and more complex boundary and initial value conditions.
Model reduction for fluid flow simulation continues to be of great interest across a number of scientific and engineering fields. Here, we explore the use of Neural Ordinary Differential Equations, a recently introduced family of continuous-depth, differentiable networks (Chen et al 2018), as a way to propagate latent-space dynamics in reduced order models. We compare their behavior with two classical non-intrusive methods based on proper orthogonal decomposition and radial basis function interpolation as well as dynamic mode decomposition. The test problems we consider include incompressible flow around a cylinder as well as real-world applications of shallow water hydrodynamics in riverine and estuarine systems. Our findings indicate that Neural ODEs provide an elegant framework for stable and accurate evolution of latent-space dynamics with a promising potential of extrapolatory predictions. However, in order to facilitate their widespread adoption for large-scale systems, significant effort needs to be directed at accelerating their training times. This will enable a more comprehensive exploration of the hyperparameter space for building generalizable Neural ODE approximations over a wide range of system dynamics.
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.
We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, a generative model that can train by maximum likelihood, without partitioning or ordering the data dimensions. For training, we show how to scalably backpropagate through any ODE solver, without access to its internal operations. This allows end-to-end training of ODEs within larger models.
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