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Physically-Constrained Data-Driven, Filtered Reduced Order Modeling of Fluid Flows
M. Mohebujjaman, L. G. Rebholz, T. Iliescu
TL;DR
The paper addresses limitations in reduced-order modeling, including closure challenges and deteriorating DDF-ROM accuracy at low ROM dimensions. It introduces a physically constrained DDF-ROM whose operators mimic fluid-equation laws, and finds it significantly more accurate than the original DDF-ROM across reproductive and predictive regimes.
Problem
Reduced-order models face closure challenges, while DDF-ROM accuracy worsens when the ROM dimension falls below a threshold.
Method
The CDDF-ROM replaces unconstrained data-driven modeling with constrained modeling that enforces energy conservation for the nonlinear operator and dissipation for the closure term.
Results
The CDDF-ROM dramatically outperformed the DDF-ROM in cross-validation and was significantly more accurate in both reproductive and predictive regimes.
Takeaways & Limitations
Physical constraints provide a more physically accurate ROM closure model than the original DDF-ROM in the investigated flow simulations.
Takeaways & Limitations
CDDF-ROM performance is sensitive to the tolerance, projection-space dimension, and constrained least-squares parameter, with reported values selected by trial and error.
Abstract
from arXiv · showhide
In our earlier work, we proposed a data-driven filtered reduced order model (DDF-ROM) framework for the numerical simulation of fluid flows, which can be formally written as \begin{equation*} \boxed{ \text{ DDF-ROM = Galerkin-ROM + Correction } } \end{equation*} The new DDF-ROM was constructed by using ROM spatial filtering and data-driven ROM closure modeling (for the Correction term) and was successfully tested in the numerical simulation of a 2D channel flow past a circular cylinder at Reynolds numbers $Re=100, Re=500$ and $Re=1000$. In this paper, we propose a {\it physically-constrained} DDF-ROM (CDDF-ROM), which aims at improving the physical accuracy of the DDF-ROM. The new physical constraints require that the CDDF-ROM operators satisfy the same type of physical laws (i.e., the nonlinear operator should conserve energy and the ROM closure term should be dissipative) as those satisfied by the fluid flow equations. To implement these physical constraints, in the data-driven modeling step of the DDF-ROM, we replace the unconstrained least squares problem with a constrained least squares problem. We perform a numerical investigation of the new CDDF-ROM and standard DDF-ROM for a 2D channel flow past a circular cylinder at Reynolds numbers $Re=100, Re=500$ and $Re=1000$. To this end, we consider a reproductive regime as well as a predictive (i.e., cross-validation) regime in which we use as little as $50\%$ of the original training data. The numerical investigation clearly shows that the new CDDF-ROM is significantly more accurate than the DDF-ROM in both regimes.
1. Introduction.
The paper develops CDDF-ROM from a hybrid filtered and data-driven ROM framework to address closure modeling while imposing physical constraints. It evaluates the constrained model against DDF-ROM for cylinder flow across Reynolds numbers and validation regimes.
- Existing ROM approaches: Projection ROMs use dominant spatial modes and Galerkin projection, while DD-ROMs infer operators by fitting a postulated ansatz to full-order data.The two approaches address reduced modeling through different operator-construction strategies.
- DDF-ROM framework: DDF-ROM combines classical projection modeling for linear operators with data-driven modeling for nonlinear operators and the closure correction.Its construction uses spatial filtering to identify the correction formula before fitting its approximation from available data.
- DDF-ROM framework: DDF-ROM is more robust to noise than standard DD-ROMs because data-driven inference is restricted to the correction term rather than all ROM operators.This design retains the Galerkin method at the framework’s core.
- Prior evaluation: DDF-ROM was significantly more accurate than the standard projection ROM for 2D cylinder flow at Re = 100, Re = 500, and Re = 1000, with similar computational costs.Both ROM costs were orders of magnitude lower than the full-order model cost.
- CDDF-ROM motivation: CDDF-ROM addresses worsening DDF-ROM results at reduced ROM dimensions by requiring the data-driven closure operators to obey physical constraints.The correction operator is required to be dissipative, while the nonlinear operator is required to conserve energy.
- Contribution: The paper presents CDDF-ROM as the first physically constrained ROM closure model and investigates it numerically after outlining the model and its evaluation.The paper’s numerical investigation and conclusions are organized in Sections 4 and 5.
2. Data-Driven Filtered ROM (DDF-ROM).
DDF-ROM filters the projected fluid model to expose missing subfilter information, then learns a correction from full-order snapshots. The resulting hybrid model is computationally efficient and can approach full-order accuracy when the true stress is available.
- ROM basis: POD constructs a reduced space from snapshots, and the velocity is represented using time-varying coefficients in that basis.The paper notes that other bases, such as DMD, could also be used.
- Spatial filtering: ROM spatial filtering identifies the correction associated with the subfilter-scale stress tensor, using the ROM projection as the filter in this paper.The framework could also accommodate other spatial filters, including the ROM differential filter.
- Projection and Galerkin ROM: The ROM projection maps a full-order state into the reduced space, while Galerkin projection produces the reduced coefficient dynamics.The reduced operators are assembled during the offline stage.
- Closure problem: The filtered ROM is an r-dimensional surrogate for the N-dimensional full-order model, but it is unclosed because its stress tensor depends on unavailable full-order information.Closure requires finding a formula approximating the stress as a function of ROM coefficients.
- Data-driven closure: DDF-ROM fits correction operators by minimizing the Euclidean discrepancy between full-order stress data and the ansatz evaluated on snapshot-derived ROM coefficients.The fitted operators are inserted into the filtered ROM to produce the closed DDF-ROM.
- Ideal DDF-ROM: With the true stress, the ideal DDF-ROM is almost as accurate as the full-order model even with r = 4; the practical ansatz loses some accuracy.The ideal model is illustrative only because it requires training data during use.
3. Physically Constrained DDF-ROM (CDDF-ROM).
CDDF-ROM imposes physical-law-inspired constraints on the data-driven correction operators. It implements these constraints by replacing unconstrained least squares with constrained least squares in the closure step.
- Physical constraints: The Galerkin linear operator is negative semidefinite, while the skew-symmetric nonlinear operator conserves energy.These properties motivate analogous constraints for the data-driven operators.
- Physical constraints: The paper asks whether the DDF-ROM correction operators should satisfy constraints consistent with those of the ROM subfilter-scale stress tensor.The stress tensor’s role is described as dissipating energy.
- Constraint formulation: Because the correction operators generate quadratic and cubic terms, CDDF-ROM replaces the direct condition with easier-to-implement constraints resembling the Galerkin operator constraints.Sufficient coefficient conditions are then given for enforcing those constraints.
- Constraint implementation: Specific sufficient conditions constrain the entries of the data-driven linear and nonlinear operators to enforce the required dissipativity and energy-conservation properties.The listed conditions include skew relations for the linear operator and index-symmetry constraints for the nonlinear operator.
- CDDF-ROM construction: The implementation replaces the unconstrained DDF-ROM least-squares problem with a constrained least-squares problem while retaining the hybrid projection/data-driven structure.The constrained operators are learned within the closure model rather than replacing the Galerkin operators.
4. Numerical Results.
The numerical study compares CDDF-ROM and DDF-ROM for two-dimensional flow past a circular cylinder. It examines accuracy in simulations at multiple Reynolds numbers and predictive cross-validation settings.
- Experimental design: The study compares constrained and unconstrained data-driven closure modeling for 2D flow past a circular cylinder at Re = 100, Re = 500, and Re = 1000.The comparison is designed to test whether constrained modeling improves accuracy.
- Experimental design: Cross-validation tests the ROMs on data not used to train the closure model, directly assessing their predictive capabilities.The paper distinguishes this predictive regime from the reproductive regime.
- Flow configuration: The benchmark uses a 2.2 × 0.41 rectangular channel containing a radius=0.05 cylinder centered at (0.2, 0.2), with no-slip walls and prescribed inflow.The flow starts from rest, uses ν = 10^-3, and has no forcing.
4.1. Test Problem Setup.
The test problem is a 2D channel flow past a circular cylinder, simulated with finite elements and time snapshots collected after the flow reaches a statistically steady state.
- Numerical setup: Snapshots are computed using linearized BDF2 in time and Scott–Vogelius finite elements in space.The first time step uses backward Euler, with Δt = 0.002 throughout.
- Numerical setup: The simulations continue to T = 17 after reaching a statistically steady state at approximately T = 5.Snapshots are taken from solutions between T = 7 and T = 7.332.
- ROM basis construction: The ROM modes are generated from snapshot averages and dominant modes obtained from an eigenvalue problem applied to average-subtracted snapshots.The singular values of the snapshot matrix are plotted in Figure 4.2.
4.2. Snapshot and ROM Generation.
The ROMs are initialized by projecting the finite-element solution into the ROM space, tested at multiple dimensions, and implemented with constrained operator fitting parameters.
- ROM initialization: The ROM initial condition at T = 6.998 is the L2 projection of the finite-element solution into the ROM space.The ROM simulations then use the corresponding initial condition at T = 7.
- Snapshot basis: Figure 4.2 plots singular values against index for the flow past a cylinder at Re = 100.
- ROM dimensions: The ROMs are tested with r = 3, r = 4, and r = 6.
- Constrained operator fitting: The constrained optimization enforces ˜A_ii ≤ −ϵ with ϵ ≥ 0 instead of requiring ˜A_ii ≤ 0.The constrained least-squares problem is solved with MATLAB lsqlin using an interior-point method.
4.3. Computational Efficiency.
Although CDDF-ROM and DDF-ROM are more accurate than G-ROM, their offline operator calculations can be costly; reduced-rank approximations balance accuracy and efficiency.
- Computational cost: CDDF-ROM and DDF-ROM are more accurate than G-ROM, but calculating ˜A and eB offline can be computationally significant.
- Cost reduction: The DDF-ROM reduces the cost of calculating ˜A and eB by approximating the correction term with lower-rank snapshot information.
- Cost reduction: The approximation replaces u_d with u_m and projects onto X_m, with r ≤ m ≤ d chosen to balance accuracy and efficiency.In the numerical investigation, m is varied to seek the highest accuracy.
4.4. Ill-Conditioning.
The least-squares problem used to infer DDF-ROM operators can be ill-conditioned, motivating truncated SVD as a remedy.
- Ill-conditioning: The least-squares problem for computing the DDF-ROM operators ˜A and eB was observed to be ill-conditioned.The paper notes that ill-conditioning also occurs in other data-driven least-squares problems.
- Ill-conditioning: Algorithm 1 uses truncated singular value decomposition to remedy this ill-conditioning.
4.5. CDDF-ROM vs DDF-ROM.
Across Re = 100, Re = 500, and Re = 1000, the CDDF-ROM is compared with the DDF-ROM and FOM using energy and, where shown, lift and drag evolutions. The CDDF-ROM is generally more accurate, especially at lower ROM dimensions, while higher-dimensional cases show less visible improvement.
- The study compares CDDF-ROM, DDF-ROM, and FOM for flow past a circular cylinder at Re = 100, Re = 500, and Re = 1000.
- Reynolds Number Re = 100: At Re = 100 and r = 4, CDDF-ROM is dramatically more accurate than DDF-ROM for energy and more accurate for drag, while lift evolution is similar.The comparison uses energy, lift, and drag coefficients versus time.
- Reynolds Number Re = 100: At Re = 100 and r = 6, CDDF-ROM remains significantly more accurate than DDF-ROM across the plotted energy, lift, and drag coefficients.
- The CDDF-ROM advantage is clearest for low r values, whereas higher r values can show no visible improvement over DDF-ROM.
- Reynolds Number Re = 500: At Re = 500, CDDF-ROM is dramatically more accurate than DDF-ROM for energy at r = 4 and significantly more accurate at r = 6.The r = 4 comparison also reports lift and drag evolutions similar to those at Re = 100.
- Reynolds Number Re = 1000: At Re = 1000, CDDF-ROM is dramatically more accurate than DDF-ROM for energy at r = 3 and significantly more accurate at r = 4.For r = 3, CDDF-ROM energy grows more slowly than DDF-ROM energy.
4.6. CDDF-ROM Cross-Validation: Predictive Investigation.
The cross-validation tests evaluate CDDF-ROM and DDF-ROM on predictive settings that use fewer snapshots and differ from the closure-model training setting. Across Reynolds numbers 100, 500, and 1000, CDDF-ROM remains accurate under severe snapshot reduction, whereas DDF-ROM is substantially less accurate, although both models are sensitive to parameter choices.
- Predictive investigation: Cross-validation tests predictive capability using settings different from training and fewer snapshots than earlier investigations.The study includes equally and unequally spaced snapshots; unequally spaced snapshots omit information from the final part of the period.
- Reynolds Number Re = 100: At Re = 100, CDDF-ROM performs very well with 10.24% equally spaced data, while DDF-ROM is very inaccurate.The comparison uses r = 4 and energy-coefficient evolution; m = r + 3 is the minimum value reported for accurate CDDF-ROM results.
- Reynolds Number Re = 100: At Re = 100, CDDF-ROM is dramatically more accurate than DDF-ROM when unequally spaced snapshots cover only the first 89% of the period.For this setting, m = r + 1 is the minimum value reported for accurate CDDF-ROM results.
- Reynolds Number Re = 500: At Re = 500, CDDF-ROM performs very well with 7% equally spaced data and is dramatically more accurate than DDF-ROM with snapshots from the first 50% of the period.Both settings use r = 4; the reported minimum m is r + 1 for accurate CDDF-ROM results.
- Parameter sensitivity: Both DDF-ROM and CDDF-ROM show high sensitivity to m, tol, and ϵ, so the numerical results depend strongly on parameter selection.The parameters were selected to best match ROM and full-order-model energy evolution, which also produced the best reported lift/drag evolutions.
5. Conclusions.
The paper introduces the physically constrained DDF-ROM (CDDF-ROM), whose data-driven operators mimic fluid-flow physical laws, and evaluates it against the original DDF-ROM. Across reproductive and predictive regimes, the CDDF-ROM is significantly more accurate, while remaining sensitive to constrained least-squares formulation choices.
- The CDDF-ROM constrains the nonlinear operator to conserve energy and the ROM closure term to be dissipative, replacing unconstrained data-driven modeling.These constraints are designed to increase the DDF-ROM’s physical accuracy.
- The CDDF-ROM was dramatically more accurate for energy coefficients and more accurate, though less substantially, for lift and drag coefficients.
- The CDDF-ROM dramatically outperformed the original DDF-ROM in both reproductive and predictive cross-validation regimes.The predictive regime used reduced training data relative to the original set.
- Future work will investigate parameter sensitivity and ill-conditioning in the constrained least-squares problem.The authors also propose testing weaker or statistical constraints that might reduce ill-conditioning and parameter sensitivity.