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Register a new userGrADE: A graph based data-driven solver for time-dependent nonlinear partial differential equations
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The physical world is governed by the laws of physics, often represented in form of nonlinear partial differential equations (PDEs). Unfortunately, solution of PDEs is non-trivial and often involves significant computational time. With recent developments in the field of artificial intelligence and machine learning, the solution of PDEs using neural network has emerged as a domain with huge potential. However, most of the developments in this field are based on either fully connected neural networks (FNN) or convolutional neural networks (CNN). While FNN is computationally inefficient as the number of network parameters can be potentially huge, CNN necessitates regular grid and simpler domain. In this work, we propose a novel framework referred to as the Graph Attention Differential Equation (GrADE) for solving time dependent nonlinear PDEs. The proposed approach couples FNN, graph neural network, and recently developed Neural ODE framework. The primary idea is to use graph neural network for modeling the spatial domain, and Neural ODE for modeling the temporal domain. The attention mechanism identifies important inputs/features and assign more weightage to the same; this enhances the performance of the proposed framework. Neural ODE, on the other hand, results in constant memory cost and allows trading of numerical precision for speed. We also propose depth refinement as an effective technique for training the proposed architecture in lesser time with better accuracy. The effectiveness of the proposed framework is illustrated using 1D and 2D Burgers equations. Results obtained illustrate the capability of the proposed framework in modeling PDE and its scalability to larger domains without the need for retraining.
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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.
Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PDEs is often a challenging task. Inspired by the traditional finite difference and finite elements methods and emerging advancements in machine learning, we propose a sequence deep learning framework called Neural-PDE, which allows to automatically learn governing rules of any time-dependent PDE system from existing data by using a bidirectional LSTM encoder, and predict the next n time steps data. One critical feature of our proposed framework is that the Neural-PDE is able to simultaneously learn and simulate the multiscale variables.We test the Neural-PDE by a range of examples from one-dimensional PDEs to a high-dimensional and nonlinear complex fluids model. The results show that the Neural-PDE is capable of learning the initial conditions, boundary conditions and differential operators without the knowledge of the specific form of a PDE system.In our experiments the Neural-PDE can efficiently extract the dynamics within 20 epochs training, and produces accurate predictions. Furthermore, unlike the traditional machine learning approaches in learning PDE such as CNN and MLP which require vast parameters for model precision, Neural-PDE shares parameters across all time steps, thus considerably reduces the computational complexity and leads to a fast learning algorithm.
Discovering the underlying behavior of complex systems is an important topic in many science and engineering disciplines. In this paper, we propose a novel neural network framework, finite difference neural networks (FDNet), to learn partial differential equations from data. Specifically, our proposed finite difference inspired network is designed to learn the underlying governing partial differential equations from trajectory data, and to iteratively estimate the future dynamical behavior using only a few trainable parameters. We illustrate the performance (predictive power) of our framework on the heat equation, with and without noise and/or forcing, and compare our results to the Forward Euler method. Moreover, we show the advantages of using a Hessian-Free Trust Region method to train the network.
Deep kernel processes (DKPs) generalise Bayesian neural networks, but do not require us to represent either features or weights. Instead, at each hidden layer they represent and optimize a flexible kernel. Here, we develop a Newton-like method for DKPs that converges in around 10 steps, exploiting matrix solvers initially developed in the control theory literature. These are many times faster the usual gradient descent approach. We generalise to arbitrary DKP architectures, by developing kernel backprop, and algorithms for kernel autodiff. While these methods currently are not Bayesian as they give point estimates and scale poorly as they are cubic in the number of datapoints, we hope they will form the basis of a new class of much more efficient approaches to optimizing deep nonlinear function approximators.
Given multiple images that describe chaotic reaction-diffusion dynamics, parameters of a PDE model are estimated using autosynchronization, where parameters are controlled by synchronization of the model to the observed data. A two-component system of predator-prey reaction-diffusion PDEs is used with spatially dependent parameters to benchmark the methods described. Applications to modelling the ecological habitat of marine plankton blooms by nonlinear data assimilation through remote sensing is discussed.
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