arxiv: 2605.11524 · v1 · submitted 2026-05-12 · 💻 cs.LG · cs.CE

Recognition: 2 theorem links

· Lean Theorem

EqOD: Symmetry-Informed Stability Selection for PDE Identification

Gnankan Landry Regis N'guessan , Bum Jun Kim

Authors on Pith no claims yet

Pith reviewed 2026-05-13 01:26 UTC · model grok-4.3

classification 💻 cs.LG cs.CE

keywords PDE identificationsparse regressionGalilean invariancestability selectionsymmetry reductiondata-driven discoverynoisy trajectory data

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The pith

EqOD detects Galilean invariance in trajectory data to shrink the candidate operator library before stability selection recovers the true PDE even under high noise.

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

The paper introduces EqOD to solve the library-size dilemma in data-driven PDE identification. When a weak-form test finds Galilean invariance, the method drops terms that the authors prove cannot appear in the governing equation; otherwise it runs randomized LASSO stability selection. A residual fallback keeps performance from falling below the full-library baseline. On eight PDEs at four noise levels the approach reaches perfect F1 scores where standard sparse-regression tools degrade, especially at 20 percent noise. The same pipeline also succeeds on clean external benchmarks and several extended cases including two-dimensional and coupled systems.

Core claim

EqOD is a fully automatic pipeline that first applies a weak-form structural test to decide whether the data exhibit Galilean invariance; if so it substitutes a provably reduced library that excludes impossible terms, otherwise it performs randomized LASSO stability selection, and in either case a residual check prevents degradation below the full library. Across thirty-two controlled experiments this yields F1 equal to 1.000 on the heat equation at 20 percent noise and wins or ties every comparison against WF-LASSO, official PySINDy 2.0, and a WSINDy reimplementation.

What carries the argument

Galilean library reduction triggered by a weak-form invariance detector, combined with randomized LASSO stability selection that prunes the space of differential operators while controlling false positives.

If this is right

Accurate recovery of the governing equation without manual library pruning for any autonomous PDE that satisfies the symmetry test.
Consistent outperformance over full-library sparse regression on both clean and noisy data across multiple equation families.
Automatic fallback that guarantees the method is never worse than the unreduced baseline.
Successful extension to two-dimensional, nonlinear-Schrödinger, coupled, and wake-flow problems without changing the core pipeline.

Where Pith is reading between the lines

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

The same symmetry-first reduction strategy could be applied to other invariances such as rotational or scaling symmetries once corresponding structural tests are developed.
Real sensor data from fluid or material experiments would be a natural next testbed where the noise-robustness gain matters most.
Integration with physics-informed neural networks could use the reduced library as a prior, further tightening the discovery loop.
Formal finite-sample guarantees for the stability-selection step under the correlated design matrices typical of PDE libraries remain open and would strengthen the method.

Load-bearing premise

The weak-form test must correctly identify Galilean invariance from finite noisy trajectories, and the true equation must obey the autonomy and library assumptions used in the exclusion proof.

What would settle it

A Galilean-invariant PDE at 20 percent noise on which the invariance detector returns negative and the subsequent model either misses a true term or includes a provably excluded term.

Figures

Figures reproduced from arXiv: 2605.11524 by Bum Jun Kim, Gnankan Landry Regis N'guessan.

**Figure 2.** Figure 2: Library reduction by EqOD pathway. The standard library has 10 candidate operators. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Solution snapshots for representative PDEs: the 1D Heat equation, Burgers shock, KdV soli [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: F1 scores as a function of noise level for Heat, Burgers, and KdV across 10 seeds. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: Identified and true coefficients for 6 representative PDEs on clean data. Hatched gray bars [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗

**Figure 6.** Figure 6: F1 at 10% noise for all 4 methods across all 8 PDEs. EqOD in blue matches or exceeds all [PITH_FULL_IMAGE:figures/full_fig_p032_6.png] view at source ↗

**Figure 7.** Figure 7: Computational scaling: wall time as a function of grid size on log-log axes. Both EqOD [PITH_FULL_IMAGE:figures/full_fig_p035_7.png] view at source ↗

**Figure 8.** Figure 8: Effect of library size on identification accuracy for Burgers at 5% noise. WF-LASSO [PITH_FULL_IMAGE:figures/full_fig_p036_8.png] view at source ↗

**Figure 9.** Figure 9: NLS identification. The left plot shows soliton modulus [PITH_FULL_IMAGE:figures/full_fig_p038_9.png] view at source ↗

**Figure 10.** Figure 10: F1 heatmap across all 8 PDEs and 4 noise levels for WF-LASSO, PySINDy, WSINDy, [PITH_FULL_IMAGE:figures/full_fig_p040_10.png] view at source ↗

**Figure 11.** Figure 11: Stability selection probability profiles. The left plot shows Heat at 10% noise, and the [PITH_FULL_IMAGE:figures/full_fig_p043_11.png] view at source ↗

**Figure 12.** Figure 12: F1 as a function of noise for all 8 PDEs with EqOD and three baselines. Curves show mean [PITH_FULL_IMAGE:figures/full_fig_p043_12.png] view at source ↗

**Figure 13.** Figure 13: 2D PDE solution fields for 2D Heat at t = 0, 2D advection-diffusion at tmid, and 2D NS vorticity at tmid. Two-dimensional extension data [PITH_FULL_IMAGE:figures/full_fig_p043_13.png] view at source ↗

**Figure 14.** Figure 14: Coupled system solutions. FHN uses u as activator and v as inhibitor, while LV uses u as prey and v as predator. Spatiotemporal patterns show the reaction-diffusion dynamics. 0 1 2 3 4 5 6 x 0.0 0.1 0.2 0.3 0.4 0.5 t Observed ut 0 1 2 3 4 5 6 x 0.0 0.1 0.2 0.3 0.4 0.5 t Predicted ut (EqOD) 0 1 2 3 4 5 6 x 0.0 0.1 0.2 0.3 0.4 0.5 t Residual 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.… view at source ↗

**Figure 15.** Figure 15: Reconstruction comparison for Burgers on clean data. The finite-difference estimate of [PITH_FULL_IMAGE:figures/full_fig_p044_15.png] view at source ↗

**Figure 16.** Figure 16: Library size selected by EqOD across noise levels. Galilean PDEs, namely Burgers, KdV, [PITH_FULL_IMAGE:figures/full_fig_p045_16.png] view at source ↗

read the original abstract

Data-driven identification of partial differential equations (PDEs) relies on sparse regression over a candidate library of differential operators, where larger libraries inflate false positives under observation noise and smaller libraries risk missing true terms. We introduce Equivariant Operator Discovery (EqOD), a fully automatic method combining two library reduction mechanisms. When Galilean invariance is detected from trajectory data via a weak-form structural test, EqOD uses the symmetry-reduced library, eliminating terms that our Galilean exclusion result proves to be absent from the governing equation. Otherwise, it applies randomized LASSO stability selection guided by classical false-positive bounds. A residual-based fallback prevents degradation below the full-library baseline. On 8 PDEs at 4 noise levels, EqOD attains $F_1 = 1.000 \pm 0.000$ on Heat at $20\%$ noise, where WF-LASSO obtains $0.475 \pm 0.181$, official PySINDy 2.0 obtains $0.000$, and the WSINDy reimplementation obtains $0.789$. Under the strict criterion that the mean F1 difference exceeds the larger of the two standard deviations, EqOD wins 7 of 32 cells. WF-LASSO wins none, and the remaining 25 cells are ties. Across all 32 cells, EqOD outperforms PySINDy 2.0.0 in 23 of 32 cells, and all 5 PySINDy wins occur on reaction PDEs. External validation on WeakIdent and PINN-SR datasets gives $F_1 = 1.000$ on all 5 clean benchmarks. NLS, 2D, coupled-system, and cylinder-wake extensions are reported. The Galilean library reduction is proved under explicit autonomy and library assumptions. The stability-selection step is motivated by classical false-positive bounds, while formal guarantees for correlated PDE design matrices remain open.

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.

Desk Editor's Note private letter to a colleague

EqOD gets some F1 gains on noisy PDEs through symmetry-based library reduction, but the gains likely depend on the weak-form test performing well at high noise.

read the letter

The punchline is that EqOD improves term selection in noisy PDE discovery by checking for Galilean invariance to shrink the candidate library, with reported F1 wins in some high-noise cases, though the advantage depends on the check working well. The paper introduces a combination of a proved Galilean exclusion rule, a weak-form structural test to detect the symmetry from trajectory data, randomized LASSO stability selection, and a residual-based fallback. This is new relative to the cited baselines. The proof eliminates absent terms under autonomy and library assumptions, which helps avoid false positives when the symmetry holds. Empirically, it achieves F1 of 1.000 on the heat equation at 20% noise where the others are lower, and strict wins in 7 of 32 cells across the tests. It also hits perfect scores on external clean benchmarks and includes extensions to other PDE types. What works is the practical fallback that prevents degradation and the inclusion of stability selection motivated by classical bounds. The external validation adds some credibility. The main soft spot is the lack of detailed evaluation for the symmetry detection itself. The stress-test concern holds: at high noise, the perfect scores require the test to correctly trigger the reduced library most of the time. Without reported precision, recall, or failure cases for the test across the 8 PDEs and noise levels, it's hard to separate the contribution of the symmetry step from the stability selection and fallback. The note about open guarantees for correlated matrices is fair but secondary. This paper is for researchers working on data-driven PDE identification in scientific machine learning. Readers extending methods like SINDy would get value from the specific integration and the head-to-head numbers. It shows honest engagement with the literature and has enough structure to merit peer review. I recommend sending it for review.

Referee Report

2 major / 2 minor

Summary. The paper introduces Equivariant Operator Discovery (EqOD) for data-driven PDE identification from noisy trajectories. It detects Galilean invariance via a weak-form structural test on data and, when detected, applies a symmetry-reduced candidate library whose exclusion of absent terms is proved under autonomy and library assumptions; otherwise it falls back to randomized LASSO stability selection with classical false-positive bounds and a residual-based safeguard. Experiments on 8 PDEs at 4 noise levels report F1 scores superior to WF-LASSO, PySINDy 2.0, and a WSINDy reimplementation (including F1=1.000 on Heat at 20% noise), with 7 strict wins under a mean-difference criterion, plus external validation on clean benchmarks and extensions to NLS, 2D, coupled, and wake flows.

Significance. If the weak-form test reliably identifies Galilean invariance from finite noisy data, EqOD offers a principled way to prune libraries using physical symmetries, reducing false positives in sparse regression while preserving completeness via the fallback. The explicit proof of term exclusion and the robustness mechanism are clear strengths that could improve reproducibility and accuracy in equation discovery pipelines.

major comments (2)

[§3.2 and Table 2] §3.2 (weak-form structural test) and Table 2 (Heat 20% noise row): the reported F1=1.000±0.000 (vs. 0.475 WF-LASSO, 0.000 PySINDy) is load-bearing for the symmetry-reduction claim, yet no precision/recall, detection-failure rates, or ablation (with vs. without the test) are supplied across the 8 PDEs and 4 noise levels; without these it is unclear whether the 7 strict wins arise from successful library reduction or from the fallback and stability-selection components.
[§2.3] §2.3 (Galilean exclusion result): the proof eliminates absent terms only under explicit autonomy and library assumptions; the manuscript does not state how these assumptions are verified or relaxed for the experimental PDEs (e.g., Heat, NLS) when the test is applied to 20% noisy finite trajectories.

minor comments (2)

[Abstract and §4] Abstract and §4: the strict win criterion (mean F1 difference exceeds the larger standard deviation) and the definition of ties should be restated explicitly when reporting the 7/32 and 23/32 counts.
[§5] §5 (extensions): the NLS, 2D, coupled-system, and cylinder-wake results are mentioned only briefly; adding a short table or paragraph on how the structural test and fallback behaved on these cases would improve completeness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and for highlighting the potential of EqOD's symmetry-informed approach. We address each major comment below with clarifications and commitments to strengthen the presentation of results and assumptions.

read point-by-point responses

Referee: [§3.2 and Table 2] §3.2 (weak-form structural test) and Table 2 (Heat 20% noise row): the reported F1=1.000±0.000 (vs. 0.475 WF-LASSO, 0.000 PySINDy) is load-bearing for the symmetry-reduction claim, yet no precision/recall, detection-failure rates, or ablation (with vs. without the test) are supplied across the 8 PDEs and 4 noise levels; without these it is unclear whether the 7 strict wins arise from successful library reduction or from the fallback and stability-selection components.

Authors: We agree that additional diagnostics would strengthen the claim that symmetry reduction, rather than the fallback mechanism, drives the reported gains. The manuscript states that the weak-form test triggers the reduced library when Galilean invariance is detected and otherwise falls back to randomized LASSO stability selection with the residual safeguard; the F1=1.000 result on Heat at 20% noise occurs under successful detection. However, we do not currently report per-PDE detection success rates, precision/recall of the test, or an explicit ablation. In the revision we will add a supplementary table listing detection outcomes across all 8 PDEs and 4 noise levels, together with an ablation comparing EqOD against its stability-selection-only variant. This will make the source of the 7 strict wins transparent. revision: yes
Referee: [§2.3] §2.3 (Galilean exclusion result): the proof eliminates absent terms only under explicit autonomy and library assumptions; the manuscript does not state how these assumptions are verified or relaxed for the experimental PDEs (e.g., Heat, NLS) when the test is applied to 20% noisy finite trajectories.

Authors: The Galilean exclusion theorem is stated under the explicit assumptions of autonomy (no explicit time dependence) and a library that is closed under the relevant differential operations. All PDEs in the experimental suite (Heat, NLS, Burgers, etc.) are autonomous, and the candidate libraries are constructed to contain all monomials up to the maximum order appearing in the true equation, satisfying the library assumption. The weak-form test is applied directly to the observed trajectories; because the test itself is derived from the weak form of the invariance condition, it does not require exact verification of the assumptions on noisy data. We will revise §2.3 to state these facts explicitly and to note that the assumptions hold for the autonomous PDE classes considered, while acknowledging that the proof does not cover non-autonomous or library-incomplete cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The Galilean reduction is presented as a proved result under explicit autonomy and library assumptions rather than a fitted or self-referential construction. Stability selection is motivated by classical false-positive bounds external to the paper. No equations or steps reduce by construction to their own inputs, fitted parameters renamed as predictions, or load-bearing self-citations. Empirical F1 comparisons are reported directly from experiments on benchmark PDEs and do not depend on tautological renaming or ansatz smuggling. The weak-form detection test is an assumption whose validity is separate from circularity analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two domain assumptions for the symmetry reduction and standard statistical motivation for stability selection; no free parameters or invented entities are introduced.

axioms (2)

domain assumption The governing PDE is autonomous
Invoked in the proof of the Galilean library reduction result.
domain assumption The candidate library satisfies closure properties under Galilean transformations
Required for the exclusion of terms proved absent from the equation.

pith-pipeline@v0.9.0 · 5658 in / 1325 out tokens · 48331 ms · 2026-05-13T01:26:56.502770+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Stability-selected support estimation. We use randomized LASSO stability selection as a non-Galilean library-pruning gate.

What do these tags mean?

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Hao Xu, Haibin Chang, and Dongxiao Zhang. Dl-pde: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data.arXiv preprint arXiv:1908.04463, 2019

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Hao Xu, Haibin Chang, and Dongxiao Zhang. DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm.J. Comput. Phys., 418:109584, 2020

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Hao Xu, Dongxiao Zhang, and Nanzhe Wang. Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data.J. Comput. Phys., 445: 110592, 2021

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Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion.Research, 6:0147, 2023

Hao Xu, Junsheng Zeng, and Dongxiao Zhang. Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion.Research, 6:0147, 2023

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Generative Adversarial Symmetry Discovery

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Symmetry-Informed Governing Equation Discovery

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Sheng Zhang and Guang Lin. Robust data-driven discovery of governing physical laws with error bars.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 474(2217), 2018

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Schaeffer [38] and Rudy et al

introduced the SINDy framework for ordinary differential equation discovery, using STLSQ. Schaeffer [38] and Rudy et al. [37] extended sparse-regression ideas to PDEs, computing spatial derivatives and applying sparse optimization over candidate libraries. These methods are effective on clean data but degrade under noise because pointwise derivatives ampl...

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use ensemble bagging to estimate inclusion probabilities under low-data and high-noise regimes. Unified sparse-optimization frameworks broaden the optimizer family available for parsimonious model discovery [10]. de Silva et al. [13] provide the PySINDy library, and the later comprehensive PySINDy release adds PDE, implicit, integral, constrained, and ens...

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The WF-LASSO baseline is EqOD without Stages 1 and 2, so it does not use symmetry detection or stability selection and isolates the contribution of library reduction. PySINDy 2.0.0 baseline.We use the official pip package pysindy==2.0.0, installed via pip install pysindy==2.0.0 . The setup uses the STLSQ optimizer with the best threshold from a grid searc...

work page arXiv 2070