An artificial neural network framework for reduced order modeling of transient flows

Omer San; Romit Maulik; Mansoor Ahmed

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An artificial neural network framework for reduced order modeling of transient flows

Omer San, Romit Maulik, Mansoor Ahmed

arXiv:1802.09474v2Feb 26, 2018physics.flu-dynphysics.comp-ph

TL;DR

The paper addresses non-intrusive reduced-order modeling for transient flows whose control parameters vary, avoiding the computational and operator-access demands of full-order and Galerkin-projection approaches. It projects simulation data onto POD modes and trains ANN architectures to evolve modal coefficients, finding that POD-ANN models—especially the residual network—perform accurately across interpolation and extrapolation tests and outperform POD-GP in the Burgers benchmark.

Problem
Transient nonlinear-flow reduction needs accurate parameter-varying predictions, while projection-based approaches can require full-order operators and may be limited for complex, highly non-stationary systems.
Method
The framework projects full-order data onto POD modes and trains sequential and residual ANN architectures to evolve reduced modal coefficients without Galerkin projection.
Results
POD-ANN consistently outperforms POD-GP in Burgers-equation tests, including interpolation and extrapolation, with POD-ANN-RN more stable and accurate beyond the training range.
Takeaways & Limitations
POD-ANN is presented as a viable non-intrusive reduced-order tool for transient nonlinear dynamics, including predictions beyond the training dataset.
Takeaways & Limitations
The study uses standard Bayesian regularization, and the effects of alternative training methodologies remain future work.

Abstract

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This paper proposes a supervised machine learning framework for the non-intrusive model order reduction of unsteady fluid flows to provide accurate predictions of non-stationary state variables when the control parameter values vary. Our approach utilizes a training process from full-order scale direct numerical simulation data projected on proper orthogonal decomposition (POD) modes to achieve an artificial neural network (ANN) model with reduced memory requirements. This data-driven ANN framework allows for a nonlinear time evolution of the modal coefficients without performing a Galerkin projection. Our POD-ANN framework can thus be considered an equation-free approach for latent space dynamics evolution of nonlinear transient systems and can be applied to a wide range of physical and engineering applications. Within this framework we introduce two architectures, namely sequential network (SN) and residual network (RN), to train the trajectory of modal coefficients. We perform a systematic analysis of the performance of the proposed reduced order modeling approaches on prediction of a nonlinear wave-propagation problem governed by the viscous Burgers equation, a simplified prototype setting for transient flows. We find that the POD-ANN-RN yields stable and accurate results for test problems assessed both within inside and outside of the database range and performs significantly better than the standard intrusive Galerkin projection model. Our results show that the proposed framework provides a non-intrusive alternative to the evolution of transient physics in a POD basis spanned space, and can be used as a robust predictive model order reduction tool for nonlinear dynamical systems.

1. Introduction

The paper motivates non-intrusive reduced-order modeling for transient, nonlinear flows because conventional POD-Galerkin models can struggle with strongly non-stationary dynamics and require access to full-order operators. It proposes POD-ANN architectures that learn modal-coefficient evolution directly from data and evaluates them against POD-GP on Burgers-equation tests.

Full-resolution engineering-flow simulations remain computationally infeasible for many practical purposes, motivating reduced-order modeling.
POD-Galerkin models evolve truncated POD coefficients through projected ordinary differential equations with low memory and computational costs.
POD-Galerkin methods can be challenging for highly non-stationary, convective, and nonlinear problems because fixed modes and truncation lose information.
The study develops a single-layer ANN approach that bypasses Galerkin projection through sequential and residual network architectures.
The authors restrict their comparison to POD-GP and note that broader comparisons with other reduced-order techniques remain future work.
Using nonlinear viscous Burgers flow, the study compares POD-ANN with POD-GP and examines interpolation, extrapolation, and architectural stability across Reynolds numbers.

2. Mathematical modeling

The study uses the viscous Burgers equation as a nonlinear convective test problem for its reduced-order modeling framework. Snapshot data are generated across ten equally spaced Reynolds numbers from 100 to 1000.

The viscous Burgers equation serves as the test case for explaining and evaluating the proposed model-reduction framework.
Re is the non-dimensional Reynolds number controlling the Burgers-equation problem.
The equation is used because it captures hallmarks of nonlinear multidimensional advection-diffusion and localized structures such as shock waves.
The database contains snapshots at 1024 spatial collocation points for ten equally spaced Reynolds numbers, Re = [100, 200, ..., 1000].
Figure 1 presents space-time Burgers solutions for representative Reynolds numbers in the database.

3. Proper orthogonal decomposition

The POD procedure builds orthonormal spatial modes from snapshot data by decomposing fluctuations, forming a correlation matrix, and eigendecomposing it. The study retains six energetic modes, capturing approximately 98.27% of the system’s energy.

Snapshot preparation: POD constructs basis functions from time-resolved field snapshots after decomposing each field into temporal mean and fluctuations.The fluctuating snapshots are used to form the correlation data matrix.
Basis construction: The snapshot correlation matrix is symmetric and non-negative, and its eigendecomposition produces orthogonal POD basis functions ranked by descending eigenvalue.Eigenvectors are normalized to enforce orthonormality between bases.
Dataset: 1010 snapshots were collected across 10 Reynolds-number realizations, with 101 snapshots between t = 0 and t = 1 for each realization.The realizations use Re = [100, 200, ..., 1000].
Mode selection: 6 most energetic POD modes capture approximately 98.27% of the energy and are used for the reduced-order analyses.The RIC index measures the percentage of captured energy for different mode counts.

4. Galerkin projection methodology

Galerkin projection reduces the governing PDE to a low-dimensional dynamical system by projecting it onto the leading POD modes. For the Burgers problem, the resulting model consists of coupled ODEs with precomputed operators and a physical initial condition.

POD truncation: The projection model truncates the POD expansion to R leading modes associated with the R largest eigenvalues, where R ≪ N.The retained modes represent the reduced basis used for the projection-based ROM.
Projection: Galerkin projection multiplies the nonlinear governing equation by each basis function and integrates over the spatial domain to obtain modal equations.The resulting projection is denoted POD-GP.
Reduced operators: The POD-GP implementation uses constant, linear, and nonlinear reduced operators formed from the mean field and POD basis functions.These quantities are represented by B_k, L_ik, and N_ijk.
Time evolution: The POD-GP consists of R coupled ODEs that can be solved efficiently because its vectors, matrices, and tensors are precomputed.The study uses a third-order Runge-Kutta scheme for time integration.
Initialization: The reduced dynamical system is initialized from the physical initial condition u(x, t = 0).

5. Artificial neural networks

The section develops a non-intrusive POD-ANN framework that learns reduced-order modal dynamics from projected data, avoiding Galerkin projection. It introduces sequential and residual architectures that map current reduced-state information to future modal coefficients for field reconstruction.

ANN foundations: Artificial neural networks establish supervised nonlinear maps between input and output spaces through successive layers, weights, biases, and activation functions.The network uses one hidden layer with tan-sigmoid activation and a generally linear output layer.
POD transforms: The learning procedure projects full-order snapshots into a POD subspace, trains on modal coefficients, and decodes predicted coefficients back into full-order fields.The encoder performs a forward POD transform, while the decoder provides the inverse transform for field reconstruction.
Training procedure: The ANN is trained by minimizing target-input error to obtain weights and biases that predict target data from posterior inputs.The study uses Bayesian regularization minimization and partitions data into 70% training, 15% validation, and 15% testing subsets.
Non-intrusive model reduction: The POD-ANN framework provides an equation-free, purely data-driven alternative to POD-GP with potential computational-efficiency and model-form-uncertainty benefits.The study evaluates the proposed models against the full-order solution and standard intrusive POD-GP results.
ANN architectures: The sequential network predicts next-step modal coefficients from Reynolds number, time, and current coefficients, while the residual network learns coefficient updates.The architectures use one hidden layer, M neurons, and R energetic POD modes; the residual update is interpreted as a discrete Euler step.
ANN architectures: The architecture illustration uses one control parameter, Re, one hidden layer with M neurons, and R most energetic POD modes.The two architectures are labeled sequential network M1 and residual network M2.

6. Numerical methods

The numerical methods section describes compact finite-difference schemes for spatial derivatives, a third-order TVD Runge–Kutta time integrator, and Simpson’s rule for evaluating inner products.

Spatial discretization: Compact difference schemes approximate both linear and nonlinear spatial differential operators for the POD-GP calculations.They are selected for high accuracy and low computational cost in fluid-dynamics problems.
Spatial discretization: The first-derivative discretization forms an α-family of tridiagonal schemes, including explicit and classical fourth-order Padé choices.α = 0 gives an explicit non-compact fourth-order scheme, while α = 1/4 yields a compact fourth-order Padé scheme.
Boundary treatment: High-order one-sided derivative formulas complete the tridiagonal systems at Dirichlet boundaries for first- and second-order derivatives.The second-derivative scheme also recovers the classical fourth-order Padé formulation at α = 1/10.
Time integration: Time integration uses an optimal third-order total variation diminishing Runge–Kutta scheme.The discrete spatial operator includes both linear and nonlinear terms.
Numerical integration: Simpson’s 1/3 rule evaluates the inner-product integrals over the spatial domain using the grid-point representation.The total number of grid points is chosen consistently with the integration rule.

7. Results

The POD-ANN models provide stable, accurate modal-coefficient predictions across interpolative and extrapolative Reynolds numbers, with the residual network outperforming the sequential network and POD-GP. The approach is non-intrusive and avoids solving a time-discrete reduced system.

Experimental setup: The assessment compares POD-GP, POD-ANN-SN, and POD-ANN-RN using six POD modes for Burgers-flow predictions.The tests use ten hidden-layer neurons and Reynolds numbers inside and beyond the training range.
Interpolative test: At Re = 750, POD-ANN preserves state accuracy longer than POD-GP, with negligible higher-mode amplitude variation relative to POD-GP’s phase and amplitude loss.The test is interpolative because training snapshots span Re = 100 to Re = 1000.
Extrapolative tests: At Re = 1250, both POD-ANN architectures remain stable and accurate, while POD-GP deviates considerably from the reference solution.POD-ANN captures the amplitudes of the first six modes, although the highest modes show slight phase variation.
Extrapolative tests: At Re = 1250, POD-ANN-RN provides more accurate estimates than POD-ANN-SN, which still captures amplitude and phase behavior reasonably well.This comparison concerns extrapolation beyond the Reynolds-number range used for training.
Extrapolative tests: At Re = 2000, POD-ANN-RN is more stable and accurate than POD-ANN-SN, while both POD-ANN approaches are observed to be two orders of magnitude faster than POD-GP.Both approaches have running times below one second with six retained modes, and the measured computational performance is dominated by data-writing for post-processing.
Interpretation: The POD-ANN framework bypasses Galerkin projection and discrete reduced-system time stepping, providing a purely data-driven alternative to POD-GP.The paper presents the residual-network architecture as a robust surrogate without requiring access to the governing equations.
Interpretation: Black-box ANN models do not retain the readily provable numerical stability, convergence, and energy-conservation properties available from first-principles models.This is identified as a limitation of replacing intrusive or first-principles reduced models with ANN surrogates.

8. Concluding remarks

The study develops an equation-free, data-driven POD-ANN framework that predicts modal evolution without explicit time stepping and compares sequential and residual architectures against POD-GP. Across Burgers-equation tests, POD-ANN supports interpolation and extrapolation, while the residual formulation is more stable; the approach remains constrained by training choices, interpretability, and scope.

The study develops an equation-free ANN framework that predicts POD-mode evolution through data-driven learning without explicit time stepping.
POD-ANN outperforms POD-GP for the Burgers equation, especially in highly truncated systems represented by only a few modes.
POD-ANN accurately interpolates and extrapolates coefficient evolution across different Reynolds numbers beyond the training datasets.
The residual network formulation is significantly more stable than the sequential network formulation, particularly for extrapolatory parameter ranges.
POD-ANN is suited to incrementally updated nonlinear modeling when interpretability is not a major concern.
The study identifies limitations including dependence on training methodology, reduced interpretability, restricted generalizability across physical regimes, and a need for broader comparisons.