Abstract
We present a novel reduced order model (ROM) approach for parameterized time-dependent PDEs based on modern learning. The ROM is suitable for multi-query problems and is nonintrusive. It is divided into two distinct stages: a nonlinear dimensionality reduction stage that handles the spatially distributed degrees of freedom based on convolutional autoencoders, and a parameterized time-stepping stage based on memory aware neural networks (NNs), specifically causal convolutional and long short-term memory NNs. Strategies to ensure generalization and stability are discussed. To show the variety of problems the ROM can handle, the methodology is demonstrated on the advection equation, and the flow past a cylinder problem modeled by the incompressible Navier–Stokes equations.
Highlights
Simulations based on first-principles models often form an essential element for understanding, designing, and optimizing problems in, for example, physics, engineering, chemistry, and economics
We present and compare two modern time series forecasting architectures, Long Short-Term Memory (LSTM) networks [20], and Causal Convolutional Neural Networks (CCNNs) [21]
To assess the performance measure the error on Ntest test trajectories for parameters values, {μ1, . . . , μNtest }, that the neural networks (NNs) have not seen in the training phase. we measure the mean relative error (MRE) at every time step and take the mean over multiple runs of the test cases: MRE(unh(μi), unh(μi))
Summary
Simulations based on first-principles models often form an essential element for understanding, designing, and optimizing problems in, for example, physics, engineering, chemistry, and economics. With an increasing complexity of the mathematical models under consideration, it is not always possible to achieve the desired fidelity of such simulations in a satisfactory time frame. High-performance computing may be costly; the improvements due to high-order discretization strongly depend on the smoothness of solutions at hand, and iterative methods are highly dependent on being able to identify suitable preconditioners. These approaches may suffer from the curse of dimensionality. ROM, a relatively recent research area, is an interesting alternative to the other approaches
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