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arxiv: 2606.08956 · v2 · pith:ZNTZBPPCnew · submitted 2026-06-08 · 💻 cs.LG

From inverse problems to neural operators: prediction, mechanism, and generalization of data-driven models

Conor Rowan This is my paper

Pith reviewed 2026-06-27 17:01 UTC · model grok-4.3

classification 💻 cs.LG
keywords scientific machine learningneural operatorsinverse problemsmechanism discoverygeneralizationdifferential equationsmodel classSINDy
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The pith

Models for physical systems generalize only when they can discover underlying mechanisms from parsimonious differential equations.

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

The paper examines different data-driven modeling approaches for physical systems, ranging from traditional inverse problems to modern neural operators. It proposes that these approaches differ primarily in the model class they assume for the input-output mapping. Drawing on ideas from philosophy of science about mechanisms, it argues that physical data stems from parsimonious differential equations, meaning only models that can recover such equations will discover mechanisms and generalize effectively. This unification helps clarify the appropriate use cases for each strategy in scientific machine learning.

Core claim

All modeling strategies can be seen as defining an input-output relation with different assumed model classes. Since data from physical systems comes from solutions to parsimonious differential equations, only those models whose class allows recovery of such equations are capable of mechanism discovery and therefore generalization.

What carries the argument

The assumed model class of the input-output relation, which determines whether the model can identify parsimonious differential equations.

If this is right

  • Sparse Identification of Nonlinear Dynamics learns sparse linear combinations of library terms and can therefore recover mechanisms.
  • Neural Ordinary Differential Equations construct the governing equation inside a neural network and can therefore recover mechanisms.
  • Neural operators that map inputs directly to outputs without differential equation structure cannot recover mechanisms.
  • Only the first two classes will generalize by discovering mechanisms rather than fitting specific input-output pairs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The unification suggests benchmarks should test generalization on systems with known parsimonious equations to separate mechanism models from others.
  • It implies that hybrid approaches combining operator learning with differential equation constraints could balance prediction accuracy and mechanism recovery.
  • The argument extends naturally to other domains where input-output data is suspected to arise from low-complexity governing rules.
  • One could test whether forcing models to output parsimonious equations improves out-of-distribution performance on real sensor data.

Load-bearing premise

Data from physical systems arises from solutions to parsimonious differential equations.

What would settle it

A physical system where a direct neural operator mapping generalizes better than a sparse differential equation model on new inputs, despite the system obeying a parsimonious equation.

Figures

Figures reproduced from arXiv: 2606.08956 by Conor Rowan.

Figure 1
Figure 1. Figure 1: A schematic of a system of interest and its experimental characterization. We take the goal of model building [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A set of 9 Laplace eigenfunctions on the space potato domain which can be used to discretize the forcing and state fields. Discretizing the input and output spaces also allow for a finite dimensional representation of the model. where F ∈ R B provides the discrete representation of the input forcing, and U ∈ R B the discrete representation of the state. Typically, basis functions are chosen such that a lar… view at source ↗
Figure 3
Figure 3. Figure 3: A set of algebraic relations appearing in the Feynman lectures on physics. Adapted from [42]. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A set of nonlinear PDEs commonly encountered in the engineering literature. Adapted from [41]. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The training region is defined to contain a specified fraction of the training inputs, which are drawn from a [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A data-driven model may fail to predict the true state field when extrapolating. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: When discussing generalization of a data-driven, authors in mainstream machine learning (as opposed to [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: In machine learning-based classification problems, the generalization error of the model is determined [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: For each training region size, 10 neural networks with random initialization are trained to match data taken noiselessly from the data generating process. Regardless of the size of the training region, the networks tend to predict different extrapolation behavior, even while agreeing in the training region. This suggests that the model is non-identifiable given the training data, which we take to rule out … view at source ↗
Figure 10
Figure 10. Figure 10: The slope and intercept of a linear model are identifiable with sufficient training data, regardless of whether [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Because the model class of the inverse problem is equivalent to the data generating process, the fit model [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: While the model class of the SINDy parameterization is more flexible than the inverse problem, the [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Based on our assumption that PDEs governing physical systems are simple, a neural network model of the [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Because the model class of PDEs involves nonlinear solves, the neural network representation of the [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: In the scientific machine learning literature, modeling strategies are taxonomized by the availability of [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: We argue that the concept of structure is a more precise way to taxonomize different data-driven modeling [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Calling PDEs which are both parsimonious and simple “scientific,” we note that coefficient maps arising [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The spreading of dye in a fluid medium is an example of a continuum system governed by the conservation [PITH_FULL_IMAGE:figures/full_fig_p029_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: A volume Ω with boundary ∂Ω immersed in a fluid medium. We are interested in the dynamics of the mixing of a dye in the fluid medium. This process can be modeled with the mass density ρ(x, t) of the dye. where q is called the “mass flux.” Passing the time derivative inside the volume integral and using the divergence theorem on the right-hand side, we can shrink the volume Ω down to a single point to obta… view at source ↗
read the original abstract

Scientists have historically relied on mathematical models based on differential equations to relate system inputs -- forces, fluxes, or heat sources -- to outputs, such as displacement, velocity, concentration, and temperature. These models rely on deep domain knowledge to determine the form of the governing differential equation, which is then calibrated with data by solving an inverse problem. In recent years, the field of Scientific Machine Learning has introduced a variety of alternative modeling strategies for physical systems. A method called Sparse Identification of Nonlinear Dynamics learns the governing equation as a sparse linear combination of terms in a user-defined library. Neural Ordinary Differential Equations construct the governing equation by taking in the state and its derivatives at the input layer of a neural network. Entirely foregoing the modeling framework of differential equations, neural operators directly learn a non-linear mapping between the system inputs and outputs. From inverse problems to neural operators, all of these modeling strategies can be conceptualized as data-driven machinery to predict a system's response over a range of inputs. It is then natural to wonder how exactly these various strategies relate to each other, and whether they can be neatly taxonomized. Drawing from the philosophical literature on scientific models, we argue that many model types have a common structure, differing only in the assumed model class of the input-output relation they define. Connecting to philosophical ideas on mechanism, and arguing that data from physical systems arises from solutions to parsimonious differential equations, we propose that only certain models are capable of mechanism discovery, and thus generalization. Our analysis is intended to unite apparently disparate modeling strategies and provide insight into their appropriate use cases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript offers a conceptual framework for relating different data-driven modeling strategies for physical systems, from traditional inverse problems using differential equations to modern approaches like Sparse Identification of Nonlinear Dynamics (SINDy), Neural ODEs, and neural operators. It argues that these methods differ primarily in the model class they assume for the input-output mapping. Drawing on philosophy of science, particularly ideas on mechanism, and the premise that physical data arises from parsimonious differential equations, the paper proposes that only models capable of discovering such mechanisms will generalize effectively. The analysis aims to unify these strategies and inform their appropriate use.

Significance. If the proposed taxonomy and the link to mechanism discovery hold, this could provide valuable insight into why certain SciML models generalize better, helping practitioners choose models based on their ability to capture underlying mechanisms rather than just fitting data. The conceptual unification is a strength, though the absence of new mathematical results or empirical validation means the significance is primarily in organizing existing ideas rather than advancing technical capabilities.

major comments (2)
  1. Abstract: The assertion that 'data from physical systems arises from solutions to parsimonious differential equations' is presented as a foundational premise for the claim about mechanism discovery and generalization, yet no supporting argument, references to specific literature in philosophy of science, or illustrative examples from physical systems are provided to establish this as a general property.
  2. Abstract: The distinction between models capable of 'mechanism discovery' and others is central to the generalization claim, but the manuscript does not provide a precise definition or operational criterion for what constitutes mechanism discovery in the context of these data-driven models, leaving the proposal open to circular reasoning where the premise defines the conclusion.
minor comments (1)
  1. The abstract mentions specific methods like SINDy and Neural ODEs but does not elaborate on how the taxonomy applies to each; expanding this with concrete mappings in the main body would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report. We address each major comment point by point below.

read point-by-point responses

  1. Referee: Abstract: The assertion that 'data from physical systems arises from solutions to parsimonious differential equations' is presented as a foundational premise for the claim about mechanism discovery and generalization, yet no supporting argument, references to specific literature in philosophy of science, or illustrative examples from physical systems are provided to establish this as a general property.

    Authors: We agree that the abstract states this premise without elaboration or citations. The full manuscript draws on the standard view in physics that governing laws take the form of parsimonious differential equations, but we did not supply explicit references or examples in the abstract itself. We will revise the abstract and the opening of the introduction to include targeted citations from the philosophy of science literature on mechanistic explanation (e.g., works by Machamer, Darden, and Craver) together with brief, concrete illustrations such as the heat equation and Newton's second law. revision: yes

  2. Referee: Abstract: The distinction between models capable of 'mechanism discovery' and others is central to the generalization claim, but the manuscript does not provide a precise definition or operational criterion for what constitutes mechanism discovery in the context of these data-driven models, leaving the proposal open to circular reasoning where the premise defines the conclusion.

    Authors: The manuscript links mechanism discovery to the recovery of an underlying sparse differential-equation structure, informed by philosophical accounts of mechanism. We acknowledge, however, that the abstract does not supply an explicit operational criterion, which could invite the circularity concern raised. In revision we will add a short clarifying paragraph (or subsection) that defines mechanism discovery operationally as the identification of a parsimonious library term or neural representation whose functional form matches the known governing equation of the system, and we will contrast this explicitly with pure input-output approximation that does not recover such structure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; conceptual argument from external philosophical premise

full rationale

The paper advances a taxonomy of modeling strategies (inverse problems, SINDy, Neural ODEs, neural operators) and a philosophical claim that only mechanism-discovering models generalize because physical data arises from parsimonious DEs. This premise is presented as drawn from the philosophy of science literature rather than derived within the paper or from the models themselves. No equations, fitted parameters, uniqueness theorems, or self-citations are invoked as load-bearing steps whose validity reduces to the target claim by construction. The argument is therefore an interpretive stance whose validity stands or falls on acceptance of the external premise, not on internal self-reference or renaming of results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that physical data arises from parsimonious differential equations and on philosophical ideas about scientific models and mechanisms.

axioms (1)
  • domain assumption Data from physical systems arises from solutions to parsimonious differential equations
    Invoked in the abstract as the basis for claiming that only certain models enable mechanism discovery and generalization.

pith-pipeline@v0.9.1-grok · 5812 in / 1109 out tokens · 23746 ms · 2026-06-27T17:01:39.757780+00:00 · methodology

discussion (0)

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Reference graph

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