Traian Iliescu

We propose, analyze, and investigate numerically a novel two‐level Galerkin reduced order model (2L‐ROM) for the efficient and accurate numerical simulation of nonlinear problems. In the first step, a relatively low‐dimensional nonlinear system is solved. In the second step, the given nonlinear equations are linearized around the solution found in the first step, and the linearized problem is solved. Although, the new 2L‐ROM can be used for general nonlinear equations, in our theoretical investigation, we consider the steady Navier–Stokes equations. We prove that, for appropriate parameters, the 2L‐ROM error bound is of the same order as the error bound for the standard 1L‐ROM. Furthermore, we develop 2L‐ROM parameter scalings that balance the 2L‐ROM's numerical accuracy and computational efficiency. In our numerical investigation, we consider the Burgers equation, which is a simplified computational setting that is often used in the ROM assessment. Our numerical investigation shows that the new 2L‐ROM significantly decreases (by a factor of 2 and even close to 3) the 1L‐ROM computational cost, without compromising its numerical accuracy. Furthermore, we show that the 2L‐ROM parameters that yield the best numerical results are similar to those obtained from the theoretical scalings.

The term foundation model is rapidly entering scientific machine learning, yet its meaning in computational science remains inconsistent. Models are increasingly labeled foundational without evidence of broad generalization across physical systems, geometries, boundary/initial conditions, tasks, or operating regimes. We propose a definition of foundation models tailored to computational science and identify core and desirable characteristics, drawing inspiration from classical foundational methods (FEM/FVM/FDM) that achieve wide reuse through problem-independent structure and principled enforcement of governing equations. We argue that scientific foundation models face unique barriers relative to language and vision models, including field-valued data, mesh and geometry dependence, heterogeneous physics, expensive adaptation, limited benchmarks, and strict requirements for interpretability and physical consistency. To make claims measurable, we introduce a reviewer- and author-friendly checklist for reporting training breadth, generalization axes, adaptation cost, extrapolation protocols, and scientific consistency checks. We also highlight a constructive hybrid path: local learning with global physics enforcement.

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.

Spatial filters have played a central role in large eddy simulation for many decades 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 or closure strategies.To answer these critical questions, we propose a novel strategy that is fundamentally different from current filter-based stabilizations and closures: We replace the traditional spatial filters with a novel data-driven stabilization operator (StabOp) that yields the most accurate results for a given resolution, quantity of interest, and stabilization strategy. Although the new StabOp could be used for both classical discretizations and ROMs, and for different types of filter-based stabilization or closure, for clarity, we investigate it for ROMs and the Leray ROM (L-ROM) stabilization. To build the new StabOp, we first postulate its model form to be a linear mapping, a quadratic mapping, or a nonlinear mapping (through a neural network), and then solve a PDE-constrained optimization problem to minimize the given loss function. Using the resulting StabOp in the given L-ROM yields a new stabilized ROM, StabOp-L-ROM. To assess the new StabOp-L-ROM, we compare it with the L-ROM and the standard ROM in the numerical simulation of four convection-dominated flows in the under-resolved regime: 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 demonstrate that the new StabOp-L-ROM can be orders-of-magnitude more accurate than the classical L-ROM tuned with the optimal filter radius in the predictive regime. Furthermore, while the new StabOp smooths the input flow fields, its smoothing mechanism is entirely different from those of classical spatial filters.

The numerical analysis of closures and stabilizations for reduced order models (ROMs) of convection-dominated flows 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, show an a priori error bound, and show parameter scalings for the TR-ROM. Our numerical investigation shows that the theoretical convergence rate and the parameter scalings with respect to the 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 the filter radius is also observed in the predictive regime.

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.