Differential equations parameterized by neural networks become expensive to solve numerically as training progresses. We propose a remedy that encourages learned dynamics to be easier to solve. Specifically, we introduce a differentiable surrogate for the time cost of standard numerical solvers, using higher-order derivatives of solution trajectories. These derivatives are efficient to compute with Taylor-mode automatic differentiation. Optimizing this additional objective trades model performance against the time cost of solving the learned dynamics. We demonstrate our approach by training substantially faster, while nearly as accurate, models in supervised classification, density estimation, and time-series modelling tasks.
Information Technology University（信息技术大学）；J. Mack Robinson College of Business（J. Mack Robinson商学院）；Georgia State University（佐治亚州立大学）
Recently, there has been a lot of interest in using neural networks for solving partial differential equations. A number of neural network-based partial differential equation solvers have been formulated which provide performances equivalent, and in some cases even superior, to classical solvers. However, these neural solvers, in general, need to be retrained each time the initial conditions or the domain of the partial differential equation changes. In this work, we posit the problem of approximating the solution of a fixed partial differential equation for any arbitrary initial conditions as…
Deep neural networks are powerful machines for visual pattern recognition, but reasoning tasks that are easy for humans may still be difficult for neural models. Humans possess the ability to extrapolate reasoning strategies learned on simple problems to solve harder examples, often by thinking for longer. For example, a person who has learned to solve small mazes can easily extend the very same search techniques to solve much larger mazes by spending more time. In computers, this behavior is often achieved through the use of algorithms, which scale to arbitrarily hard problem instances at the…
This paper investigates the use of artificial neural networks (ANNs) to solve differential equations (DEs) and the construction of the loss function which meets both differential equation and its initial/boundary condition of a certain DE. In section 2, the loss function is generalized to $n^\text{th}$ order ordinary differential equation(ODE). Other methods of construction are examined in Section 3 and applied to three different models to assess their effectiveness.
University of Ontario Institute of Technology（安大略理工大学）
We introduce a technique based on the singular vector canonical correlation analysis (SVCCA) for measuring the generality of neural network layers across a continuously-parametrized set of tasks. We illustrate this method by studying generality in neural networks trained to solve parametrized boundary value problems based on the Poisson partial differential equation. We find that the first hidden layer is general, and that deeper layers are successively more specific. Next, we validate our method against an existing technique that measures layer generality using transfer learning experiments. …
NewPINNs: Physics-Informing Neural Networks Using Conventional Solvers for Partial Differential Equations
Department of Mathematics, University of California Riverside（加州大学河滨分校数学系）；Department of Mechanical Engineering, University of California Riverside（加州大学河滨分校机械工程系）；German Innovation Center, Procter & Gamble Service GmbH（普罗cter与 gamble 服务德国创新中心）
We present Elvet, a Python package for solving differential equations and variational problems using machine learning methods. Elvet can deal with any system of coupled ordinary or partial differential equations with arbitrary initial and boundary conditions. It can also minimize any functional that depends on a collection of functions of several variables while imposing constraints on them. The solution to any of these problems is represented as a neural network trained to produce the desired function.
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）
A conventional approach to train neural ordinary differential equations (ODEs) is to fix an ODE solver and then learn the neural network's weights to optimize a target loss function. However, such an approach is tailored for a specific discretization method and its properties, which may not be optimal for the selected application and yield the overfitting to the given solver. In our paper, we investigate how the variability in solvers' space can improve neural ODEs performance. We consider a family of Runge-Kutta methods that are parameterized by no more than two scalar variables. Based on the…
Helmholtz-Zentrum Dresden-Rossendorf（德累斯顿-罗森多夫亥姆霍兹中心）；Technische Universität Dresden（德累斯顿工业大学）；Center for Advanced Systems Understanding (CASUS)（高级系统理解中心）
Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. However, recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. Scanning the parameters of the underlying model significantly increases the runtime as the simulations have to be cold-started for each parameter configuration. Machine Learning based surrogate models denote promising ways for learning complex relationship among input, parameter and solution…
Memorial Sloan Kettering Cancer Center（纪念斯隆凯特琳癌症中心）；Indian Institute of Science Education and Research, Tirupati（印度科学教育与研究学院蒂鲁帕蒂分校）；Indian Institute of Technology, Madras（印度理工学院马德拉斯分校）
Purpose: We seek to use neural networks (NNs) to solve a well-known system of differential equations describing the balance between T cells and HIV viral burden. Materials and Methods: In this paper, we employ a 3-input parallel NN to approximate solutions for the system of first-order ordinary differential equations describing the above biochemical relationship. Results: The numerical results obtained by the NN are very similar to a host of numerical approximations from the literature. Conclusion: We have demonstrated use of NN integration of a well-known and medically important system of fir…
Neural Network Differential Equation (NN DE) solvers have surged in popularity due to a combination of factors: computational advances making their optimization more tractable, their capacity to handle high dimensional problems, easy interpret-ability of their models, etc. However, almost all NN DE solvers suffer from a fundamental limitation: they are trained using loss functions that depend only implicitly on the error associated with the estimate. As such, validation and error analysis of solution estimates requires knowledge of the true solution. Indeed, if the true solution is unknown, we…
We explore in detail a method to solve ordinary differential equations using feedforward neural networks. We prove a specific loss function, which does not require knowledge of the exact solution, to be a suitable standard metric to evaluate neural networks' performance. Neural networks are shown to be proficient at approximating continuous solutions within their training domains. We illustrate neural networks' ability to outperform traditional standard numerical techniques. Training is thoroughly examined and three universal phases are found: (i) a prior tangent adjustment, (ii) a curvature f…
Indian Institute of Technology Kanpur（印度理工学院坎普尔分校）；Julia Computing；Massachusetts Institute of Technology（麻省理工学院）；University of Maryland Baltimore（马里兰大学巴尔的摩分校）；Pumas AI
Democratization of machine learning requires architectures that automatically adapt to new problems. Neural Differential Equations (NDEs) have emerged as a popular modeling framework by removing the need for ML practitioners to choose the number of layers in a recurrent model. While we can control the computational cost by choosing the number of layers in standard architectures, in NDEs the number of neural network evaluations for a forward pass can depend on the number of steps of the adaptive ODE solver. But, can we force the NDE to learn the version with the least steps while not increasing…
University of Alaska Fairbanks（阿拉斯加大学费尔班克斯分校）；Oak Ridge National Laboratory（橡树岭国家实验室）
We propose a novel deep learning (DL) approach to solve one-dimensional non-linear elliptic, parabolic, and hyperbolic problems on graphs. A system of physics-informed neural network (PINN) models is used to solve the differential equations, by assigning each PINN model to a specific edge of the graph. Kirkhoff-Neumann (KN) nodal conditions are imposed in a weak form by adding a penalization term to the training loss function. Through the penalization term that imposes the KN conditions, PINN models associated with edges that share a node coordinate with each other to ensure continuity of the …
We motivate the use of neural networks for the construction of numerical solutions to differential equations. We prove that there exists a feed-forward neural network that can arbitrarily minimise an objective function that is zero at the solution of Poisson's equation, allowing us to guarantee that neural network solution estimates can get arbitrarily close to the exact solutions. We also show how these estimates can be appreciably enhanced through various strategies, in particular through the construction of error correction networks, for which we propose a general method. We conclude by pro…
