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REVIEW 4 major objections 5 minor 6 cited by

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

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This position paper argues that 'foundation model' needs a rigorous definition in computational science, and presents the Data-Driven Finite Element Method as a framework that satisfies it.

desk verdict A useful position paper with a clear definition of foundation models for computational science, but the DD-FEM evidence is self-referential and the 'arbitrarily large' extrapolation claim outruns the data. read the letter →

arxiv 2505.22904 v2 pith:YK26FRWI submitted 2025-05-28 cs.LG cs.AIcs.NAmath.NA

Youngsoo Choi , Siu Wun Cheung , Youngkyu Kim , Ping-Hsuan Tsai , Alejandro N. Diaz , Ivan Zanardi , Seung Whan Chung , Dylan Matthew Copeland
show 4 more authors
Coleman Kendrick William Anderson Traian Iliescu Matthias Heinkenschloss
This is my paper · ORCID
classification cs.LGcs.AIcs.NAmath.NA MSC 65N3068T07
keywords foundationmodelscomputationalsciencescientificmachinelearningdata-drivenfiniteelementmethoddomaindecompositiongeneralizationreducedorderpositionpaper
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This position paper contends that the term 'foundation model' is being applied to computational science too loosely: many neural surrogates carry the label while generalizing only within a narrow problem class. It proposes a formal definition — a foundation model in computational science is a data-driven model trained on a broad distribution of scientific application types or physical systems that generalizes across problems, computational domains, tasks, and physical conditions without retraining from scratch or structural modification, and serves as a reusable base. To show the definition is attainable, the paper introduces the Data-Driven Finite Element Method (DD-FEM), which trains local basis functions on small subdomains and then assembles them into arbitrarily large global domains, solving the governing equations to enforce physics. A comparison table holds DD-FEM to the paper's own criteria and marks it as satisfying all of them, where existing neural-operator and transformer-based 'foundation models' fall short on at least one; if the definition is adopted, the term would carry an enforceable checklist for authors and reviewers.

What carries the argument

The load-bearing object is the definition itself, made operational by a checklist of essential and desirable characteristics: data-driven learning, training on a broad distribution of systems, wide generalization without retraining or structural modification, reusability, and the desirable properties of scientific consistency, robust space-time extrapolation, data scalability, and mathematical rigor. The framework offered as a concrete realization is DD-FEM, whose mechanism is local learning with global assembly: basis functions are pretrained on small subdomain problems, then stitched into a global domain, where solving the discretized governing equations (via static condensation, residual minimization with continuity constraints, or discontinuous Galerkin flux coupling) enforces physics and provides the structural basis for consistency, stability, and error analysis. This mechanism is what decouples data generation cost from global problem scale and is what the paper claims earns DD-FEM its foundation-model status.

What would settle it

Train data-driven elements only on small subdomain problems, then assemble them to solve a global problem whose physics couples distant subdomains in a way no local training configuration contains — for instance a shock wave or a long-range pressure field crossing many element boundaries. If the relative error against a full-order monolithic solve climbs well above the roughly 1–4% range reported in Section 5, or if elements must be retrained to hold that accuracy, the transfer assumption that carries DD-FEM's foundation-model claim fails.

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Extended reading notes

Core claim

The paper's central claim is the definition in Section 2.2: a foundation model in computational science is a data-driven model trained on a broad distribution of scientific application types or physical systems, which exhibits wide generalization across scientific problems, computational domains, tasks, and physical conditions without requiring retraining from scratch or structural modification, and serves as a reusable base. The companion claim is that the Data-Driven Finite Element Method (DD-FEM) is a step toward realizing such models and satisfies the definition's key criteria: local data-driven basis functions are trained on small subdomain simulations, assembled into a global domain of any size constructible from those elements, and the governing equations are then solved to enforce physical laws. In Table 2 the paper marks DD-FEM as satisfying every criterion — training distribution, cross-domain generalization, reusability, space-time extrapolation, data scalability, scientific consistency, and mathematical rigor — while marking the surveyed neural-operator and transformer-based models as only partially satisfying or missing at least one criterion. The numerical support comes from component-ROM experiments on lattice elasticity, porous-media Navier–Stokes flow, and Burgers flow, reporting speedups over full-order solvers with relative errors around 1–4%.

Load-bearing premise

The central claim rests on a transfer assumption: basis functions trained on small subdomains must stay accurate when assembled into much larger, never-seen global problems without retraining, and the paper's evidence for that is a few examples drawn from the authors' own earlier work.

Editorial extensions

If this is right

  • Authors and reviewers gain a shared checklist: a model may be called a foundation model in computational science only if it demonstrates training on a broad distribution, generalization without retraining, reusability, and ideally scientific consistency and mathematical rigor.
  • If DD-FEM's local-to-global transfer holds, the data bottleneck that blocks scientific foundation models relaxes, because diverse training data can come from cheap small-subdomain simulations rather than huge global solves.
  • A community repository of pretrained data-driven elements could let practitioners assemble solvers for new geometries, boundary conditions, or materials from existing components, with minimal or no retraining.
  • Because DD-FEM keeps the governing-equation solve, it opens a route to convergence, stability, and error-estimate guarantees, qualities the paper argues black-box surrogates lack.
  • Mixing classical polynomial elements with data-driven elements in one assembly lets existing simulation codes adopt the framework incrementally instead of all at once.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's criteria are qualitative, so an obvious extension is to quantify each one — adaptation cost as a fraction of retraining cost, relative error growth with assembled domain size, and the breadth of physics families, geometries, and boundary conditions in the training distribution — turning Table 2's checkmarks into comparable, falsifiable numbers.
  • The framework's generality implies a concrete untested experiment: whether bases trained on data from one discretization (say, finite element snapshots) remain accurate when assembled and solved in another (finite volume or finite difference), which the paper suggests but does not demonstrate.
  • If the element-library vision succeeds, the limiting resource shifts from model capacity to dataset curation; which small subdomain problems best span the space of global physics becomes a benchmark-design question the paper leaves open.
  • The paper's own question of whether a domain-specific model can qualify is likely to resolve into a graded standard rather than a yes-or-no one, since most current candidates pass the breadth test only within a single physics family.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This position paper argues that the term "foundation model" is used too loosely in computational science and proposes a formal definition tailored to that field (Section 2.2), together with essential and desirable characteristics (Sections 2.2 and 2.3). It then introduces the Data-Driven Finite Element Method (DD-FEM), a framework that trains local data-driven basis functions on small subdomains and assembles them into a global solver, and claims that DD-FEM is "a step toward" foundation models in computational science. The claim is supported by a collection of numerical examples from prior component-ROM papers (Section 5.2), a discussion of implementation considerations (Section 7), and a comparison table of existing foundation-model claims (Section 9, Table 2). The paper is explicitly a position statement rather than a full empirical study, and the authors disclaim that their definition is the only possible one.

Significance. If the proposed definition were adopted, it could impose useful discipline on the rapidly growing literature that labels models as foundation models for science and engineering. The conceptual parallel between classical FEM modularity and modern data-driven pretraining is timely and may help frame future benchmarks. The explicit articulation of challenges in Section 3, the reproducibility guidance in Section 7, and the critical survey of existing claims in Section 9 are valuable contributions. However, the paper's central empirical assertion—that DD-FEM satisfies the paper's own foundation-model criteria—is currently supported only by retrospective examples from the authors' prior work, without code, data, or error statistics, and with several unqualified claims (e.g., "arbitrarily large systems") that outrun the evidence. Thus the paper is more persuasive as a call for clarity than as a demonstration that DD-FEM already meets the proposed bar.

major comments (4)
  1. [Section 4.1] The assertion that pretrained data-driven elements "can be assembled to solve arbitrarily large systems ... without requiring retraining from scratch" is load-bearing for the claim that DD-FEM is a foundation model, but it is not supported by the evidence in Section 5.2. The largest demonstrations are 16×16, 10×10, and 32×32 element assemblies, and no scaling law or error bound is reported as a function of the number of assembled elements. The cited error analyses (refs. 26, 28, 29) are local or static-condensation results and do not by themselves bound the error of a global assembly with unseen inter-element couplings. I recommend either presenting a rigorous error-propagation result or softening the claim to "without retraining at the scales tested here."
  2. [Section 5.2.3 / Table 2] The cross-PDE generalization example uses RBF interpolation to generate initial guesses for the Poisson test parameters, so the Poisson result is not a pure DD-FEM prediction. This detail weakens the "Cross-domain Generalization" checkmark in Table 2, and the 6% average error for Poisson should be reported with this caveat. The authors should clarify exactly what RBF interpolation contributes and whether the same claim would hold without it.
  3. [Table 2] Table 2 awards DD-FEM full checkmarks for Reusability, Space-time Extrapolation, Scientific Consistency, and Mathematical Rigor, but the criteria are defined in this paper and the evidence consists of a few favorable examples from the authors' own prior papers. This is close to circularity: the paper sets the bar and then scores its own proposed method against that bar using selected retrospective results. I recommend replacing full checkmarks with "partially satisfied" for at least Space-time Extrapolation and Mathematical Rigor, or adding a caveat that the table reflects the authors' assessment rather than an independent benchmark.
  4. [Section 5.2 / Section 7] The numerical evidence in Section 5.2 is reported from prior papers with no code or data, no error bars, and no discussion of variability across seeds or problem instances. For a paper whose stated goal is "rigor" and which itself emphasizes reproducibility in Section 7, this is a significant gap. The authors should either provide access to the underlying data and scripts, report error statistics over multiple trials, or explicitly relabel these examples as illustrative rather than as validation of the foundation-model claim.
minor comments (5)
  1. [Section 2.2] The phrase "without requiring retraining from scratch or structural modification" is not operationalized; the paper does not specify what counts as "minimal fine-tuning" or "structural modification." Without a metric, the criterion is hard to apply to other models in Table 2.
  2. [Section 4.1 / Figure 7] Figure 7 is referenced in Section 4.1 but appears only in Section 6; this makes the analogy harder to follow. The figure should be introduced where it is first cited or the citation moved.
  3. [Section 5.2.3] The sentence "radial basis function (RBF) interpolation was used to generate initial guesses at the test parameters" lacks details such as the kernel, the number of centers, and the hyperparameter values; please provide enough information for reproducibility.
  4. [Section 9] The word "seemlessly" appears in the paragraph advocating hybrid approaches; it should be "seamlessly."
  5. [Table 2] The legend uses combined symbols like "✓✗" which are difficult to parse in the rendered table; using two separate columns per criterion, or a three-level symbol with a clear legend, would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the paper defines a term, proposes DD-FEM, and illustrates it with self-cited prior numerical results; the load-bearing extrapolation claim is conditional and the authors explicitly list global error guarantees as open.

full rationale

This is a position paper, not a derivation: Section 2.2 proposes a definition, Section 4 introduces DD-FEM as a framework, and Section 5 collects previously reported component-ROM results. No numerical prediction is computed from a parameter fitted to the same quantity it predicts. The central 'arbitrarily large systems' claim in Section 4.1 is explicitly conditional ('provided the global domain can be constructed from the pretrained elements'), and Section 11 candidly lists the missing formal guarantees ('Can we extend classical FEM and domain decomposition theory to rigorously characterize convergence, stability, and error propagation in DD-FEM?'), so the claim is an open research direction rather than a conclusion imported by definition. The Section 5.2 evidence is drawn from the authors' own component-ROM papers ([29], [31], [32], [34]), which is a heavy self-citation load; however, those papers state their own training pipelines, governing equations, and error numbers (e.g., Eq. (44) of [32], Eq. (33) of [31], Algorithm 2 of [29]) and are thus externally falsifiable numerical studies rather than unverified assertions. The RBF interpolation in Section 5.2.3 is disclosed as an initial-guess generator, not as the solver output, so the Poisson result is not identical to the interpolant by construction. Table 2 is an author-assigned checklist applied to the authors' own framework using criteria defined in the same paper; that is self-assessment, not circularity, and it is tempered by the paper's own open-questions section. Overall, the central definitional content has independent conceptual content, so the paper should not be scored for circularity; the unsupported global-scaling claim is a correctness and evidence-concentration concern, not a circular one.

Assumptions & free parameters 3 free parameters · 3 assumptions · 2 invented entities

The paper introduces no new physics or mathematics; its central contributions are a definition (a community proposal) and a framework (DD-FEM) that repackages the authors' prior component ROM work. The free parameters listed are hyperparameters from the cited experiments, which are not reported here. The axioms are the standard FEM assumptions plus the validity of the authors' own prior error analyses.

free parameters (3)
  • SVD basis truncation rank = not reported
    Used in the lattice and Navier-Stokes component ROM examples (Section 5.2.1) to compress local data; the rank is a hyperparameter chosen by the authors in prior work.
  • Autoencoder latent dimension = not reported
    Used in the Burgers nonlinear manifold example (Section 5.2.1) and the cross-PDE example (Section 5.2.3).
  • RBF interpolation hyperparameters = not reported
    RBF interpolation is used to generate initial guesses for the Poisson test parameters in Section 5.2.3; kernel and width are not specified.
assumptions (3)
  • standard math Standard finite element method convergence theory (weak formulation, Galerkin projection, stability, consistency) applies.
    Invoked in Section 4.1 to argue DD-FEM can inherit mathematical rigor from FEM.
  • domain assumption Component-based reduced order model error bounds from cited works are correct.
    The paper relies on error analyses in refs [26,28,29] to support the accuracy claims for DD-FEM.
  • ad hoc to paper The proposed definition of foundation models is the appropriate one for computational science.
    The definition in Section 2.2 is a position statement, not derived from established theory, and is used to evaluate all models including the authors' own DD-FEM in Section 9.
invented entities (2)
  • Data-Driven Finite Element Method (DD-FEM)
    purpose: A framework combining local learned bases with global numerical assembly to serve as a foundation model for computational science.
    Introduced in Section 4.1 as a new framework name, but it is a synthesis of existing component ROM methods; no falsifiable predictions are made beyond the cited prior experiments.
  • Data-driven element (data-driven basis)
    purpose: Learned local basis functions trained on subdomain data, intended to replace polynomial bases in FEM.
    The concept is defined in Section 4.1 and is essentially the component ROM ideas from prior papers; no independent evidence is provided in this paper.

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Cite this review

Pith. "Pith review of Defining Foundation Models for Computational Science: A Call for Clarity and Rigor." pith.science (2026). https://pith.science/paper/YK26FRWI

@misc{pith2026250522904,
  author       = {Pith},
  title        = {Pith review of: Defining Foundation Models for Computational Science: A Call for Clarity and Rigor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YK26FRWI}},
  note         = {Machine review of arXiv:2505.22904}
}
read the original abstract

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.

Figures

Figures reproduced from arXiv: 2505.22904 by the authors.

Figure 1
Figure 1. A foundation model can consolidate knowledge from diverse problem domains into a single, [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Schematic description of the DD-FEM framework procedure: (i) Local basis construction; [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Demonstration of extrapolation in space. (a) lattice-type structure design optimization, (b) [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Demonstration of generalization to different source functions in Poisson’s equation. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Demonstration of generalization to different types of PDEs. [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Three conceptual pathways to enable DD-FEM: (i) compression-based methods (e.g., [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Analogy between NLP and DD-FEM. Beyond this comparative insight, the hybrid nature of DD-FEM also affords extensibility in both directions: • From Traditional FEM: Techniques such as adaptive mesh refinement, multigrid solvers, hp-adaptivity, domain decomposition, and …

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.