arXiv subjects

Ping-Hsuan Tsai

Publications and source records attributed to Ping-Hsuan Tsai.

9 recordsLinked to original sources

arXiv2026

Learning Long-Term Stable Operator Inference Reduced-Order Models of Fluid Flows through Online Spatial Filtering

This paper introduces an online evolve--filter--relax (EFR) strategy for long-term stability of Operator Inference (OpInf) reduced-order models (ROMs) of complex fluid flow simulations. The main novelty of the new EFR-OpInf strategy is the use of online (i.e., at the learned ROM online evaluation level) spatial filtering inspired from large eddy simulation to significantly improve long-term stability and predictive performance of standard OpInf. Furthermore, the EFR-OpInf strategy reduces, and in some cases even eliminates, the need for standard $L^2$ regularization, while providing physical interpretability for the OpInf hyperparameters. The EFR-OpInf framework is modular, readily integrated into existing OpInf workflows, and accommodates a user-selected ROM filtering strategy. We demonstrate the EFR-OpInf's effectiveness using a fully non-intrusive projection-based ROM filter, a ROM differential filter, and a hybrid projection-differential ROM filter. The new EFR-OpInf models are evaluated on a high-Péclet-number convection--diffusion--reaction problem that embeds the variation in one parameter and two unsteady Navier--Stokes problems that focus on predictions beyond a training horizon: a transitional two-dimensional flow past a cylinder and a three-dimensional turbulent minimal channel flow. Across these three scenarios, EFR-OpInf can reduce prediction errors by up to an order of magnitude relative to standard OpInf. Moreover, EFR-OpInf remains stable over long prediction horizons in cases where standard OpInf diverges. Depending on the ROM filter used, EFR-OpInf's computational cost is comparable to that of standard OpInf.

math.NA↗

arXiv2026

StabOp: A Data-Driven Stabilization Operator for Reduced Order Modeling

Spatial filters have played a central role in large eddy simulation and, more recently, in reduced order model (ROM) stabilization for convection-dominated flows. Nevertheless, important open questions remain: in under-resolved regimes, which filter is most suitable for a given stabilization or closure model? Moreover, once a filter is selected, how should its parameters, such as the filter radius, be determined? Addressing these questions is essential for the reliable design and performance of filter-based stabilization strategies. To answer these questions, we propose a novel strategy that differs fundamentally from current filter-based approaches: we replace traditional spatial filters with a data-driven stabilization operator (StabOp) that yields accurate results for a given resolution, quantity of interest, and stabilization strategy. Although the new StabOp can be used for both classical discretizations and ROMs, and for different types of filter-based stabilization or closure, for clarity we focus on ROMs and the Leray stabilization. To build the new StabOp, we postulate its model form as a linear, quadratic, or nonlinear mapping, and then solve a PDE-constrained optimization problem to minimize a given loss function. Using the resulting StabOp in the Leray ROM (L-ROM) yields a new stabilized ROM, StabOp-L-ROM. To assess the StabOp-L-ROM, we compare it with the L-ROM and the standard ROM in numerical simulations of four flows: 2D flow past a cylinder at Re=500, lid-driven cavity at Re=10000, 3D flow past a hemisphere at Re=2200, and minimal channel flow at Re=5000. Our numerical results show that the StabOp-L-ROM can be orders of magnitude more accurate than the classical L-ROM tuned with an optimal filter radius in the predictive regime. Furthermore, while the new StabOp smooths the input flow fields, its smoothing mechanism differs from that of classical spatial filters.

physics.flu-dyn↗

arXiv2025

Accelerating Galerkin Reduced-Order Models for Turbulent Flows with Tensor Decomposition

Galerkin-based reduced-order models (G-ROMs) offer efficient and accurate approximations for laminar flows but require hundreds to thousands of modes $N$ to capture the complex dynamics of turbulent flows. This makes standard G-ROMs computationally expensive due to the third-order advection tensor contraction, requiring the storage of $N^3$ entries and the computation of $2N^3$ operations per timestep. As a result, such ROMs are impractical for realistic applications like turbulent flow control. In this work, we consider problems that demand large $N$ values for accurate G-ROMs and propose a novel approach that accelerates G-ROMs by utilizing the CANDECOMP/PARAFAC (CP) tensor decomposition to approximate the advection tensor as a sum of $R$ rank-1 tensors. We also leverage the partial skew-symmetry property of the advection tensor and derive two conditions for the CP decomposition to preserve this property. Moreover, we investigate the low-rank structure of the advection tensor using singular value decomposition (SVD) and compare the performance of G-ROMs accelerated by CP (CPD-ROM) and SVD (SVD-ROM). Demonstrated on problems from 2D periodic to 3D turbulent flows, the CPD-ROM achieves at least a $10$-fold speedup and a $16.7$-fold reduction in nonlinear term evaluation costs compared to the standard G-ROM. The skew-symmetry preserving CPD-ROM demonstrates improved stability in both the reproduction and predictive regimes, and enables the use of smaller rank $R$. Singular value analysis reveals a persistent low-rank structure in the $H^1_0$-based advection tensor, and CP decomposition achieves at least an order of magnitude higher compression ratio than SVD.

physics.flu-dyn↗

arXiv2025

Defining Foundation Models for Computational Science: A Call for Clarity and Rigor

The widespread success of foundation models in natural language processing and computer vision has inspired researchers to extend the concept to scientific machine learning and computational science. However, this position paper argues that as the term "foundation model" is an evolving concept, its application in computational science is increasingly used without a universally accepted definition, potentially creating confusion and diluting its precise scientific meaning. In this paper, we address this gap by proposing a formal definition of foundation models in computational science, grounded in the core values of generality, reusability, and scalability. We articulate a set of essential and desirable characteristics that such models must exhibit, drawing parallels with traditional foundational methods, like the finite element and finite volume methods. Furthermore, we introduce the Data-Driven Finite Element Method (DD-FEM), a framework that fuses the modular structure of classical FEM with the representational power of data-driven learning. We demonstrate how DD-FEM addresses many of the key challenges in realizing foundation models for computational science, including scalability, adaptability, and physics consistency. By bridging traditional numerical methods with modern AI paradigms, this work provides a rigorous foundation for evaluating and developing novel approaches toward future foundation models in computational science.

cs.LG↗

arXiv2025

Verifiability and Limit Consistency of Eddy Viscosity Large Eddy Simulation Reduced Order Models

Large eddy simulation reduced order models (LES-ROMs) are ROMs that leverage LES ideas (e.g., filtering and closure modeling) to construct accurate and efficient ROMs for convection-dominated (e.g., turbulent) flows. Eddy viscosity (EV) ROMs (e.g., Smagorinsky ROM (S-ROM)) are LES-ROMs whose closure model consists of a diffusion-like operator in which the viscosity depends on the ROM velocity. We propose the Ladyzhenskaya ROM (L-ROM), which is a generalization of the S-ROM. Furthermore, we prove two fundamental numerical analysis results for the new L-ROM and the classical S-ROM: (i) We prove the verifiability of the L-ROM and S-ROM, i.e, that the ROM error is bounded (up to a constant) by the ROM closure error. (ii) We introduce the concept of ROM limit consistency (in a discrete sense), and prove that the L-ROM and S-ROM are limit consistent, i.e., that as the ROM dimension approaches the rank of the snapshot matrix, $d$, and the ROM lengthscale goes to zero, the ROM solution converges to the \emph{``true solution"}, i.e., the solution of the $d$-dimensional ROM. Finally, we illustrate numerically the verifiability and limit consistency of the new L-ROM and S-ROM in two under-resolved convection-dominated problems that display sharp gradients: (i) the 1D Burgers equation with a small diffusion coefficient; and (ii) the 2D lid-driven cavity flow at Reynolds number $Re=15,000$.

physics.flu-dyn↗

arXiv2025

Symbolic Regression of Data-Driven Reduced Order Model Closures for Under-Resolved, Convection-Dominated Flows

Data-driven closures correct the standard reduced order models (ROMs) to increase their accuracy in under-resolved, convection-dominated flows. There are two types of data-driven ROM closures in current use: (i) structural, with simple ansatzes (e.g., linear or quadratic); and (ii) machine learning-based, with neural network ansatzes. We propose a novel symbolic regression (SR) data-driven ROM closure strategy, which combines the advantages of current approaches and eliminates their drawbacks. As a result, the new data-driven SR closures yield ROMs that are interpretable, parsimonious, accurate, generalizable, and robust. To compare the data-driven SR-ROM closures with the structural and machine learning-based ROM closures, we consider the data-driven variational multiscale ROM framework and two under-resolved, convection-dominated test problems: the flow past a cylinder and the lid-driven cavity flow at Reynolds numbers Re = 10000, 15000, and 20000. This numerical investigation shows that the new data-driven SR-ROM closures yield more accurate and robust ROMs than the structural and machine learning ROM closures.

math.NA↗

arXiv2025

Local Reduced-Order Modeling for Electrostatic Plasmas by Physics-Informed Solution Manifold Decomposition

Despite advancements in high-performance computing and modern numerical algorithms, computational cost remains prohibitive for multi-query kinetic plasma simulations. In this work, we develop data-driven reduced-order models (ROMs) for collisionless electrostatic plasma dynamics, based on the kinetic Vlasov-Poisson equation. Our ROM approach projects the equation onto a linear subspace defined by the proper orthogonal decomposition (POD) modes. We introduce an efficient tensorial method to update the nonlinear term using a precomputed third-order tensor. We capture multiscale behavior with a minimal number of POD modes by decomposing the solution manifold into multiple time windows and creating temporally local ROMs. We consider two strategies for decomposition: one based on the physical time and the other based on the electric field energy. Applied to the 1D1V Vlasov-Poisson simulations, that is, prescribed E-field, Landau damping, and two-stream instability, we demonstrate that our ROMs accurately capture the total energy of the system both for parametric and time extrapolation cases. The temporally local ROMs are more efficient and accurate than the single ROM. In addition, in the two-stream instability case, we show that the energy-windowing reduced-order model (EW-ROM) is more efficient and accurate than the time-windowing reduced-order model (TW-ROM). With the tensorial approach, EW-ROM solves the equation approximately 90 times faster than Eulerian simulations while maintaining a maximum relative error of 7.5% for the training data and 11% for the testing data.

math.NA↗

arXiv2024

A Priori Error Bounds and Parameter Scalings for the Time Relaxation Reduced Order Model

The a priori error analysis of reduced order models (ROMs) for fluids is relatively scarce. In this paper, we take a step in this direction and conduct numerical analysis of the recently introduced time relaxation ROM (TR-ROM), which uses spatial filtering to stabilize ROMs for convection-dominated flows. Specifically, we prove stability, an a priori error bound, and parameter scalings for the TR-ROM. Our numerical investigation shows that the theoretical convergence rate and the parameter scalings with respect to ROM dimension and filter radius are recovered numerically. In addition, the parameter scaling can be used to extrapolate the time relaxation parameter to other ROM dimensions and filter radii. Moreover, the parameter scaling with respect to filter radius is also observed in the predictive regime.

math.NA↗

arXiv2023

A Time-Relaxation Reduced Order Model for the Turbulent Channel Flow

Reg-ROMs are stabilization strategies that leverage spatial filtering to alleviate the spurious numerical oscillations generally displayed by the classical G-ROM in under-resolved numerical simulations of turbulent flows. In this paper, we propose a new Reg-ROM, the time-relaxation ROM (TR-ROM), which filters the marginally resolved scales. We compare the new TR-ROM with the two other Reg-ROMs in current use, i.e., the L-ROM and the EFR-ROM, in the numerical simulation of the turbulent channel flow at $Re_τ = 180$ and $Re_τ = 395$ in both the reproduction and the predictive regimes. For each Reg-ROM, we investigate two different filters: (i) the differential filter (DF), and (ii) a new higher-order algebraic filter (HOAF). In our numerical investigation, we monitor the Reg-ROM performance for the ROM dimension, $N$, and the filter order. We also perform sensitivity studies of the three Reg-ROMs for the time interval, relaxation parameter, and filter radius. The numerical results yield the following conclusions: (i) All three Reg-ROMs are significantly more accurate than the G-ROM and (ii) more accurate than the ROM projection, representing the best theoretical approximation of the training data in the given ROM space. (iii) With the optimal parameter values, the TR-ROM is more accurate than the other two Reg-ROMs in all tests. (iv) For most $N$ values, DF yields the most accurate results for all three Reg-ROMs. (v) The optimal parameters trained in the reproduction regime are also optimal for the predictive regime for most $N$ values. (vi) All three Reg-ROMs are sensitive to the filter radius and the filter order, and the EFR-ROM and the TR-ROM are sensitive to the relaxation parameter. (vii) The optimal range for the filter radius and the effect of relaxation parameter are similar for the two $\rm Re_τ$ values.

math.NA↗