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A deep learning enabler for non-intrusive reduced order modeling of fluid flows

S. Pawar, S. M. Rahman, H. Vaddireddy, O. San, A. Rasheed, P. Vedula

arXiv:1907.04945v1physics.comp-phphysics.flu-dyn

TL;DR

The paper addresses the computational cost of high-fidelity fluid simulations and the need for reduced-order models that do not require governing-equation information. It introduces modular DNN frameworks that learn state updates, residuals, or numerical slopes from snapshot data, and demonstrates accurate predictions on dynamical systems and Boussinesq cavity flows. The results support the proposed approach as a non-intrusive predictive tool, while the study notes that its POD-based setup remains a scope boundary.

  • Problem

    High-fidelity fluid simulations are computationally expensive, while projection-based ROMs can depend on governing-equation operators that may be unavailable or difficult to extract.

  • Method

    The paper learns reduced dynamical evolution with modular DNN frameworks that predict states directly or learn residual and backward-difference numerical-slope information from state histories.

  • Results

    Across Kraichnan–Orszag, Lorenz, and Boussinesq cavity-flow examples, the DNN frameworks provide accurate predictions, outperforming Galerkin ROM in the reported Boussinesq comparisons.

  • Takeaways & Limitations

    Snapshot-based DNN forecasting can provide a non-intrusive reduced-order modeling framework for complex fluid flows without requiring prior governing-equation information.

  • Takeaways & Limitations

    The study uses snapshot POD to represent high-dimensional data in a low-dimensional space, leaving the necessity of POD preconditioning as an open question.

Abstract

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In this paper, we introduce a modular deep neural network (DNN) framework for data-driven reduced order modeling of dynamical systems relevant to fluid flows. We propose various deep neural network architectures which numerically predict evolution of dynamical systems by learning from either using discrete state or slope information of the system. Our approach has been demonstrated using both residual formula and backward difference scheme formulas. However, it can be easily generalized into many different numerical schemes as well. We give a demonstration of our framework for three examples: (i) Kraichnan-Orszag system, an illustrative coupled nonlinear ordinary differential equations, (ii) Lorenz system exhibiting chaotic behavior, and (iii) a non-intrusive model order reduction framework for the two-dimensional Boussinesq equations with a differentially heated cavity flow setup at various Rayleigh numbers. Using only snapshots of state variables at discrete time instances, our data-driven approach can be considered truly non-intrusive, since any prior information about the underlying governing equations is not required for generating the reduced order model. Our \textit{a posteriori} analysis shows that the proposed data-driven approach is remarkably accurate, and can be used as a robust predictive tool for non-intrusive model order reduction of complex fluid flows.

I. INTRODUCTION

Reduced-order modeling addresses the prohibitive cost of resolving complex flows, but projection-based methods can lose accuracy for transport-dominated dynamics and require intrusive equation information. The paper proposes a deep-neural-network framework for non-intrusive forecasting from data, bypassing Galerkin projection.

  • Full-order simulations remain computationally prohibitive because resolving flow features requires many degrees of freedom, especially across broad parameter ranges.
  • Projection-based ROMs reduce governing-equation complexity through lower-dimensional basis expansions, with POD targeting dominant energy-containing flow modes.
  • Intrusive ROMs derive reduced evolution equations through Galerkin projection, whereas non-intrusive approaches infer underlying dynamics from data without prior governing-equation information.
  • POD-Galerkin ROMs can struggle with transport-dominated phenomena because deformed structures are not accurately captured by a few fixed modes.
  • The framework is positioned within broader data-driven efforts to construct non-intrusive reduced-order models for high-dimensional dynamical systems.
  • The proposed NIROM bypasses Galerkin projection and decomposes modeling into basis representation and forecasting using deep feed-forward neural-network architectures.

II. LEARNING FRAMEWORK

The framework uses deep neural networks to advance nonlinear dynamical systems from state histories, either predicting states directly or learning residuals and numerical slopes for iterative updates. These designs support different numerical schemes and train with standard neural-network objectives.

  • Network architecture: Each DNN maps inputs through layered weighted transformations with biases and activation functions, using ReLU in hidden layers and a linear output layer.Weights are learned through backpropagation and optimization by minimizing prediction error.
  • Frameworks: The DNN frameworks predict nonlinear dynamical systems from state history supplied over p time steps.The framework includes sequential, residual, and backward-difference variants.
  • DNN-S: DNN-S learns a direct update from y(n) to y(n+1), advancing the system state sequentially during prediction.Past state history can be incorporated into the input features.
  • DNN-R: DNN-R learns the difference between consecutive states and applies the predicted residual to update the current state.The paper reports that residual learning can improve stability and prediction accuracy.
  • DNN-B: DNN-B learns a discrete numerical slope, using a second-order backward-difference scheme to update the state iteratively.The numerical slope is the predicted variable, and mean squared error can be used directly as the loss function.
  • Evaluation: Prediction quality is evaluated with total RMSE, which aggregates component-wise errors between true and predicted states across time.The metric sums the root mean square error for each dynamical-system component.

III. TIME SERIES PREDICTION FOR DYNAMICAL SYSTEMS

The paper first tests the DNN frameworks on nonlinear dynamical systems before applying them to reduced-order modeling. The preliminary examples include the three-mode Kraichnan–Orszag system and the chaotic Lorenz system.

  • Experimental setup: The preliminary experiments evaluate the DNN frameworks on nonlinear dynamical systems before their use in the non-intrusive ROM setup.This tests the forecasting capability of the frameworks independently of the fluid-flow ROM application.
  • Test systems: The examples comprise the three-mode Kraichnan–Orszag problem and the chaotic Lorenz system.The Kraichnan–Orszag and Lorenz cases are presented in separate subsections.
  • Iterative prediction: The trained outputs of the different DNN frameworks are used to update the solution in time from an initial condition.Table I summarizes the learned output associated with each framework.

A. Kraichnan-Orszag System

The Kraichnan-Orszag experiments test several DNN frameworks on a three-state nonlinear system using different input histories. All frameworks reproduce the system states accurately, with DNN-R and DNN-B outperforming DNN-S for p = 1 and near-identical predictions for p = 4.

  • Problem: The Kraichnan-Orszag system is a three-state nonlinear dynamical system whose modeling is challenging because of discontinuities at specified planes.The initial conditions use y1(0) = 1, y2(0) = 0.1ξ, y3(0) = 0 with ξ ∈ [−1, 1].
  • Setup: The numerical experiment integrates the system from t = 0 to 10 using 1000 time steps and trains all DNN frameworks with the same hyperparameters.For the reported case, ξ = 0.5 gives [y1 y2 y3]T = [1.0 0.05 0.0]T.
  • Results: For p = 1, all DNN frameworks correctly predict the three states, while DNN-R and DNN-B perform better than DNN-S.The DNN-S prediction is described as very slightly shifted from the original state.
  • Results: For p = 4, the states predicted by all DNN frameworks are almost the same as the true state.The comparison uses total RMSE for the different frameworks and numbers of neural-network inputs.

B. Lorenz System

The Lorenz experiment evaluates DNN forecasting on a highly nonlinear chaotic system. DNN-R and DNN-B track the true trajectory longer than DNN-S, while adding temporal history extends tracking for DNN-S.

  • Problem: The Lorenz system is challenging to model because of its highly nonlinear and chaotic behavior.The experiment uses α = 10, ρ = 28, and β = 8/3.
  • Setup: The neural networks are trained on a numerically generated true solution from t = 0 to 25 using the initial condition [y1 y2 y3]T = [−8 7 27]T.The time step is Δt = 0.01, corresponding to N = 2500 time steps.
  • Results: With p = 1, DNN-R and DNN-B follow the true Lorenz trajectory longer than DNN-S.Because the system is chaotic, small state-prediction errors can produce larger forecast errors.
  • Results: Increasing the input history from p = 1 to p = 4 makes DNN-S follow the true trajectory longer, whereas DNN-R and DNN-B show no similar behavior.
  • Interpretation: The Lorenz attractor’s overall dynamics should be assessed separately from exact individual-state trajectories after divergence begins.The positive Lyapunov exponent causes small initial perturbations to diverge exponentially.

IV. NON-INTRUSIVE REDUCED ORDER MODELING (NIROM)

The paper extends its DNN forecasting framework to non-intrusive reduced order modeling for real-world flow problems. The approach targets robust and efficient recovery of reduced-order dynamics without relying on intrusive projection operations.

  • Motivation: Successful predictions for the nonlinear Kraichnan-Orszag and chaotic Lorenz systems motivate testing the DNN frameworks on a real-world model-order-reduction problem.
  • Framework: The proposed NIROM framework transforms a partial differential equation system into ordinary differential equations for reduced-order modeling.Its stated goal is to recover the reduced-order dynamics of the underlying flow phenomena.
  • Framework: The framework is designed as a robust and efficient non-intrusive reduced order model using the introduced neural-network architectures.

A. NIROM framework

The NIROM framework models generalized PDE systems from discrete snapshots without access to the governing operator. It decomposes the task into basis representation and forecasting and is presented as modular and physics-agnostic.

  • Framework formulation: The generalized NIROM framework represents a PDE system in terms of a state u and an operator R containing physical-process terms.The state may include velocity, pressure, and temperature.
  • Illustrative test case: The Lorenz trajectory experiment provides a dynamical-system test case in which the neural network is trained on true-solution data from t = 0 to 25 with p = 1.
  • Assumptions: The framework assumes that the operator R(u(x, t); x, t) is unavailable and uses only discrete snapshots of u(x, t).
  • Framework design: NIROM decomposes the problem into basis representation and forecasting subproblems.The paper describes this decomposition as physics-agnostic and modular.

B. Problem definition: Boussinesq equations

The study applies a non-intrusive reduced-order modeling workflow to two-dimensional buoyancy-driven flow in a differentially heated cavity. The setup uses reduced representations, neural-network forecasting, and flow diagnostics across Rayleigh numbers ranging from periodic to chaotic behavior.

  • Problem setup: Rayleigh number is introduced as a control parameter for natural-convection heat transfer alongside the dimensionless parameters Re, Pr, and Ri.The simulations fix Pr = 0.71 and Ri = 1.
  • NIROM workflow: The NIROM workflow constructs orthonormal basis functions, encodes reduced coefficients, trains a forecasting model, and decodes predicted coefficients into fields.The workflow uses DNN forecasting and can predict reduced coefficients at arbitrary times.
  • Governing formulation: The full-order simulation uses a vorticity-streamfunction formulation, with velocity recovered from the streamfunction and a Poisson equation enforcing the divergence-free constraint.This formulation avoids numerical complexity associated with primitive variables.
  • Flow regimes: The flow is periodic at lower Rayleigh numbers but becomes non-periodic and turbulent as Rayleigh number increases.The framework is tested at Ra = 3.4 × 10^5 for periodic flow and Ra = 9.4 × 10^5 for chaotic flow.

C. NIROM framework for Boussinesq equations

The NIROM framework reduces high-fidelity Boussinesq snapshots with POD and forecasts the resulting modal coefficients using DNNs. Its main challenge is that higher-Rayleigh-number flows require many modes for adequate energy capture, while the non-intrusive approach remains accurate without physical time integration.

  • POD construction: POD bases are computed from high-fidelity snapshots by mean subtraction, correlation-matrix eigendecomposition, and descending eigenvalue ordering.The resulting modes represent the dominant spatial content of the vorticity and temperature fields.
  • POD energy content: The first 10 modes capture more than 99% of total energy at Ra = 3.4 × 10^5 and Ra = 5.4 × 10^5 for both vorticity and temperature.Eigenvalue decay is faster at lower Rayleigh numbers.
  • POD energy content: At Ra = 7.4 × 10^5 and Ra = 9.4 × 10^5, the first 10 vorticity modes capture only 76% and 69% of total energy, respectively.More than 40 modes are required to capture over 95% of the energy at higher Rayleigh numbers.
  • Encoder-decoder forecasting: The encoder maps vorticity and temperature fields to time-dependent modal coefficients, which DNN frameworks forecast before decoder reconstruction.The retained basis contains R modes with R << M, where M is the snapshot count.
  • Encoder-decoder forecasting: The DNN-based ROM is fully non-intrusive because it uses no physical equations for time integration of the solution field.The approach forecasts reduced coefficients and reconstructs the fields from the retained modes.

V. NUMERICAL RESULTS FOR BOUSSINESQ EQUATIONS

The Boussinesq cavity experiments evaluate DNN-based non-intrusive ROMs across Rayleigh numbers, temporal coefficients, temperature fields, and Nusselt-number predictions. The frameworks generally track the true reduced-order projection accurately, while performance depends on flow complexity and input history.

  • Evaluation scope: The study evaluates DNN frameworks using vorticity coefficients, final-time temperature fields, and time-averaged engineering quantities at two Rayleigh numbers.The analysis compares predictions with full-order simulations, true projections, and ROM-G.
  • Modal-coefficient prediction: At Ra = 3.4 × 10^5, DNN predictions closely follow the true projected vorticity coefficients, and increasing p improves DNN-R predictions.DNN-S and DNN-B already perform well with p = 1, while DNN-R improves with p = 4.
  • Modal-coefficient prediction: At Ra = 9.4 × 10^5, Galerkin projection becomes unbounded with fewer modes, whereas DNN predictions remain bounded but show larger deviations from the true projection.The higher-Rayleigh-number flow is chaotic and turbulent, making accurate low-dimensional Galerkin prediction more difficult.
  • Temperature-field prediction: For final-time temperature fields, DNN-S and DNN-B are close to the true projection at lower Rayleigh number, while higher-Rayleigh predictions show discrepancies near cavity boundaries.Using p = 4 produces a slight improvement in the higher-Rayleigh temperature prediction.
  • Scope and limitations: The authors caution that the networks are trained on true projected data and therefore can recover at most the true projection rather than the full-order solution.Out-of-sample predictions are less accurate than in-sample predictions, especially for higher-Rayleigh cases.
  • Engineering quantities: All DNN frameworks accurately predict time-averaged Nusselt numbers near the true projection, but higher-Rayleigh out-of-sample accuracy is less consistent.The standard deviation is predicted sufficiently accurately, while ROM-G overpredicts or becomes unbounded in several higher-Rayleigh comparisons.

VI. CONCLUDING REMARKS

The paper presents a non-intrusive DNN reduced order modeling framework that predicts dynamical-system evolution and performs accurately for benchmark systems and differentially heated cavity flows. The framework predicts temperature fields and time-averaged Nusselt numbers sufficiently accurately, while Galerkin projection becomes unbounded for higher-Rayleigh-number flows.

  • Framework: The framework uses encoder-decoder DNNs to iteratively predict ROM modal coefficients, with multiple temporal legs incorporating short-term state history.The DNN-B framework leverages backward-difference schemes and can be generalized to other numerical schemes.
  • Benchmark systems: All DNN frameworks correctly predict the Kraichnan-Orszag states and reproduce Lorenz-system dynamics through the Lorenz attractor, although not its long-duration trajectories.
  • Cavity flow: At higher Rayleigh number, modal-coefficient predictions deviate especially out of sample, but the proposed framework still predicts the temperature field and time-averaged Nusselt number accurately enough.
  • Cavity flow: For the cavity flow, non-intrusive ROM frameworks perform exceptionally well at lower Rayleigh number and predict instantaneous temperature and time-averaged Nusselt number with sufficient accuracy at higher Rayleigh number.
  • Comparison with ROM-G: Galerkin projection performs well at low Rayleigh number but becomes unbounded and produces incorrect predictions at higher Rayleigh number.Recovering accurate physics with Galerkin projection in the complex-flow case requires more modes and therefore higher computational cost.

APPENDIX: NUSSELT NUMBER CALCULATION

The appendix computes instantaneous Nusselt numbers from temperature gradients along the cavity’s left wall and then derives their mean and standard deviation over the simulation interval. The same procedure is applied to full-order and reduced-order solutions, with reduced-order fields reconstructed from retained POD modes.

  • Instantaneous Nusselt number: The instantaneous Nusselt number is evaluated from the temperature gradient along the left wall at x = 0.For full-order data, the wall gradient uses a right-sided finite-difference scheme.
  • Numerical integration: The full-order temperature-gradient integration uses the fourth-order accurate Simpson’s rule.
  • Statistics: The mean and standard deviation are computed from instantaneous Nusselt numbers over time t = 0 to t = 100.N denotes the total number of time steps in this interval.
  • Reduced-order calculation: For reduced-order models, the temperature field is reconstructed from modal coefficients using the retained POD basis before applying the Nusselt-number procedure.This study retains R = 10 POD modes.
  • Reduced-order calculation: A faster reduced-order calculation stores gradients of the mean temperature and each basis function to obtain instantaneous temperature gradients directly.
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