Source-linked AI summary
Data-Driven Deep Learning of Partial Differential Equations in Modal Space
Kailiang Wu, Dongbin Xiu
TL;DR
The paper addresses how to recover unknown time-dependent PDE dynamics from solution data without identifying the PDE’s exact terms. It approximates the evolution operator in generalized Fourier space using a residual-network-based deep model, and reports accurate prediction across several PDE examples, with Gibbs-type oscillations remaining in some global approximations.
Problem
The problem is to reconstruct unknown time-dependent PDE dynamics from solution snapshots rather than directly identifying their governing-equation terms.
Method
The method projects snapshot data into modal space and trains a residual-network-based deep neural network to approximate the finite-dimensional evolution operator for recursive prediction.
Results
The framework provides accurate prediction in the reported examples, including advection up to t = 20 and improved higher-mode accuracy over the Galerkin solver for inviscid Burgers’ equation.
Takeaways & Limitations
Approximating the evolution operator in modal space supports data-driven prediction of unknown PDEs without requiring the precise governing equation.
Takeaways & Limitations
Global approximations can produce Gibbs-type oscillations unless special treatment such as filtering is used, which the work does not pursue.
Abstract
from arXiv · showhide
We present a framework for recovering/approximating unknown time-dependent partial differential equation (PDE) using its solution data. Instead of identifying the terms in the underlying PDE, we seek to approximate the evolution operator of the underlying PDE numerically. The evolution operator of the PDE, defined in infinite-dimensional space, maps the solution from a current time to a future time and completely characterizes the solution evolution of the underlying unknown PDE. Our recovery strategy relies on approximation of the evolution operator in a properly defined modal space, i.e., generalized Fourier space, in order to reduce the problem to finite dimensions. The finite dimensional approximation is then accomplished by training a deep neural network structure, which is based on residual network (ResNet), using the given data. Error analysis is provided to illustrate the predictive accuracy of the proposed method. A set of examples of different types of PDEs, including inviscid Burgers' equation that develops discontinuity in its solution, are presented to demonstrate the effectiveness of the proposed method.
1. Introduction.
The paper develops a general framework for approximating unknown time-dependent PDEs by learning their evolution operators rather than identifying equation terms. It reduces the infinite-dimensional PDE problem to finite dimensions through modal-space representations and uses neural networks for numerical approximation.
- Existing approaches use symbolic regression or function approximation to recover physical laws, dynamical systems, or governing-equation terms.
- The paper targets a general numerical framework for learning unknown time-dependent PDEs, addressing limitations of methods focused on particular PDE types or predefined term dictionaries.
- The method recovers the PDE evolution operator, whose accurate approximation enables solution prediction and effectively characterizes the equation’s evolution.
- Because PDE systems are infinite-dimensional, the approach uses a generalized Fourier or modal space to obtain a finite-dimensional dynamical system for modal coefficients.
- The paper combines neural-network approximation with error analysis and numerical examples to assess predictive accuracy and applicability.
2. Problem Setup.
The problem is to reconstruct the dynamics of an unknown autonomous PDE from noisy solution snapshots. The reconstructed dynamics should support accurate future solution prediction while assuming known boundary conditions.
- The state variable satisfies an autonomous time-dependent PDE with unknown interior operator L, known boundary operator B, and initial condition u0.
- The spatial domain, time variable, differential operator, and boundary operator define the PDE setting, with known boundary conditions assumed throughout.
- Available training information consists of solution snapshots w(x,tj) formed from the exact solution plus acquisition noise or error.
- The central goal is to reconstruct the unknown governing dynamics from snapshots so the learned system can predict future solutions.
3. Finite Dimensional Approximation.
The infinite-dimensional evolution-operator learning problem is projected into a finite-dimensional space and represented through modal coefficients. The resulting operator is approximated and recursively applied, with error controlled by projection and approximation terms.
- The method replaces direct governing-equation identification with approximation of the evolution operator and first reduces the PDE problem from infinite to finite dimension.
- A finite-dimensional space Vn is chosen within the solution Hilbert space, with basis functions satisfying the known linear boundary conditions.
- Solutions in Vn are represented by modal coefficients through a bijective mapping between the function space and R^n.
- The finite-dimensional operator E∆,n advances an approximate solution in Vn by one time lag, providing a tractable surrogate for the unknown evolution operator.
- Recursive prediction error is bounded through projection error and accumulated finite-dimensional evolution-operator errors.
- The coefficient-space evolution operator M∆,n is equivalent to learning the finite-dimensional operator and therefore transforms infinite-dimensional operator learning into finite-dimensional dynamics learning.
4. Numerical Approach.
The method reduces unknown PDE learning to finite-dimensional evolution-operator approximation in modal space, trained from time-separated coefficient data. A neural network then advances modal coefficients iteratively, while the error analysis separates network approximation from projection error.
- Algorithm: The workflow selects a finite-dimensional basis and projection, converts solution snapshots into modal data, trains an approximation of the evolution operator, and advances it for prediction.The procedure is presented as a general framework that can use approximation methods beyond neural networks, although deep networks are suitable for relatively high-dimensional inputs.
- Data construction: Training data consist of paired modal vectors whose components are separated by a time lag Δ.For autonomous systems, only the time difference matters, and the paper assumes a constant lag after introducing possibly varying lags.
- Data construction: Snapshot-based pairing projects solution fields separated by Δ into the finite-dimensional space before learning the modal evolution operator.Pairs may come from the same trajectory, but pairs from different trajectories and many initial conditions are strongly preferred for more effective recovery.
- Data construction: Alternatively, training pairs can be generated by sampling initial modal vectors and advancing them with black-box simulation software or a device.This directly constructs initial states in the approximation space and produces corresponding solution snapshots after the chosen time lag.
- Neural-network modeling: A block ResNet models the finite-dimensional evolution operator, with parameters learned by minimizing a loss over coefficient pairs.The trained network is applied recursively to projected initial conditions to obtain approximate modal coefficients and reconstructed solution fields.
- Error analysis: The prediction error combines neural-network approximation error with projection error determined by the approximation space and solution regularity.The analysis assumes bounded evolution and trained network operators and defines projection error relative to the exact solution.
5. Numerical Examples.
The numerical examples test the modal-space ResNet approach across linear, nonlinear, noisy, multidimensional, and shock-forming PDE settings. Predictions generally agree with exact or projected solutions, while errors and Gibbs oscillations appear under long-time evolution or discontinuities.
- Experiment scope: The experiments cover linear advection, linear diffusion, viscous and inviscid Burgers’ equations, noisy data, and a two-dimensional advection-diffusion problem.The inviscid Burgers’ case produces shocks, while most tests use one-dimensional physical space and noiseless data.
- Experimental setup: The examples use global orthogonal polynomials to define the finite-dimensional modal space, with training data generated in modal space.A decay condition is imposed in some examples so higher modes remain smaller than lower modes, effectively imposing smoothness on the training data.
- Advection equation: In the advection example, the learned model produces accurate predictions through t = 20, although the relative error grows over time.The test initial condition is smooth but lies outside the approximation space Vn.
- Diffusion equation: For diffusion, the predicted solutions agree well with the exact solution, and learned expansion coefficients agree with coefficients obtained by orthogonal projection.The diffusion validation is conducted through t = 3 in the reported comparison.
- Noisy data: With ±2% and ±5% multiplicative data noise, predictions remain fairly robust, while higher noise produces relatively larger numerical errors.The noisy-data experiment trains the neural-network models for 500 epochs before comparing predictions with the exact solution.
- Inviscid Burgers’ equation: For discontinuous Burgers’ solutions, both neural-network and Galerkin predictions exhibit Gibbs-type oscillations, but the neural-network prediction is reported as visibly better than Galerkin.The neural-network method is data-driven, whereas the Galerkin comparison requires knowledge of the governing equation.
6. Conclusion.
The paper presents a data-driven framework for learning unknown time-dependent autonomous PDEs by approximately recovering their evolution operators rather than exact PDE terms. Modal-space reduction, error analysis, and varied test problems support recursive system prediction and demonstrate the method’s applicability.
- The framework trains deep neural networks, particularly residual networks, to approximate unknown PDE evolution operators instead of identifying exact equation terms.Because the evolution operator characterizes solution evolution, its recovery supports prediction of the underlying PDE.
- The method reduces infinite-dimensional operator learning to finite dimensions through modal-space, or generalized Fourier-space, representations.Error analysis is used to quantify predictive accuracy.
- Recursive use of the learned operator enables accurate system prediction across a variety of test problems.The paper presents these examples to demonstrate the method’s applicability and potential.