arxiv: 2604.18889 · v1 · submitted 2026-04-20 · 💻 cs.LG

AC-SINDy: Compositional Sparse Identification of Nonlinear Dynamics

Peter Racioppo This is my paper

Pith reviewed 2026-05-10 04:35 UTC · model grok-4.3

classification 💻 cs.LG

keywords sparse identification of nonlinear dynamicsarithmetic circuitsSINDycompositional feature constructionlatent state inferencesystem identificationchaotic systemsmulti-step supervision

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The pith

AC-SINDy builds nonlinear features for dynamics identification through compact arithmetic circuit compositions rather than explicit libraries.

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

The paper introduces AC-SINDy to extend the sparse identification of nonlinear dynamics framework by replacing large enumerated feature libraries with arithmetic circuits. These circuits construct candidate nonlinear terms via repeated compositions of linear functions and multiplicative interactions, which permits sparsity to be enforced directly on the computational graph. A second formulation separates latent state inference from shared dynamics learning and adds multi-step supervision to increase robustness against measurement noise. Experiments on nonlinear and chaotic systems indicate that the resulting models recover accurate governing equations while scaling more favorably than standard SINDy.

Core claim

AC-SINDy replaces explicit feature libraries with a structured representation based on arithmetic circuits that compose linear functions and multiplicative interactions, yielding a compact parameterization over which sparsity can be imposed directly on the graph; it further decouples state estimation from dynamics identification through latent variable inference, shared parameters, and multi-step supervision to improve noise tolerance while preserving interpretability of the recovered equations.

What carries the argument

Arithmetic circuits that generate nonlinear candidate terms by composing linear functions with multiplicative interactions, with sparsity applied over the circuit graph instead of an enumerated basis.

If this is right

The combinatorial explosion of candidate terms that limits standard SINDy in high-dimensional or strongly nonlinear regimes is avoided because the circuit parameterization grows more gradually with complexity.
Recovered equations stay directly interpretable because each multiplicative or linear step in the circuit corresponds to an explicit functional interaction.
Robustness to partial or noisy observations increases through the separation of latent state estimation from the shared dynamics model and the use of multi-step prediction losses.
The same circuit structure can in principle be reused across multiple trajectories or related systems because the dynamics parameters are shared.

Where Pith is reading between the lines

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

Physical systems whose vector fields admit low-depth arithmetic representations may be identified more reliably by this method than by exhaustive libraries.
The circuit approach could be paired with other sparse regression objectives beyond differential-equation fitting, such as discrete-time maps or partial differential equations.
Systematic comparison of required circuit depth against the complexity of recovered equations might expose structural regularities in classes of chaotic or nonlinear systems.

Load-bearing premise

The nonlinear terms needed to describe the target dynamics can be represented, to sufficient accuracy, by finite compositions of linear functions and multiplications.

What would settle it

AC-SINDy would be falsified if, on a system whose governing equations are known and recoverable by standard SINDy, the circuit-based method either misses essential terms or produces substantially larger trajectory prediction errors on held-out data.

Figures

Figures reproduced from arXiv: 2604.18889 by Peter Racioppo.

**Figure 2.** Figure 2: Parameter growth for standard SINDy (explicit enumeration) versus AC-SINDy (composi [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Training loss during iterative pruning. Loss spikes occur when parameters are removed, [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: Left: Predicted vs. true trajectory for a single state variable. Right: Phase-space trajectory. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: Noisy system identification with Gaussian noise ( [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

read the original abstract

We present AC-SINDy, a compositional extension of the Sparse Identification of Nonlinear Dynamics (SINDy) framework that replaces explicit feature libraries with a structured representation based on arithmetic circuits. Rather than enumerating candidate basis functions, the proposed approach constructs nonlinear features through compositions of linear functions and multiplicative interactions, yielding a compact and scalable parameterization and enabling sparsity to be enforced directly over the computational graph. We also introduce a formulation that separates state estimation from dynamics identification by combining latent state inference with shared dynamics and multi-step supervision, improving robustness to noise while preserving interpretability. Experiments on nonlinear and chaotic systems demonstrate that the method recovers accurate and interpretable governing equations while scaling more favorably than standard SINDy.

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

AC-SINDy replaces SINDy's explicit library with arithmetic circuits of linear maps and multiplies, plus a latent-state multi-step trick for noise, but the abstract gives no numbers to show it actually works better.

read the letter

The core new pieces are the arithmetic-circuit parameterization that builds nonlinear terms through repeated linear combinations and multiplications instead of an enumerated library, and the split between latent state inference and shared dynamics with multi-step supervision. Both are absent from the SINDy papers cited in the abstract. The circuit version should in principle let sparsity act directly on the computation graph and avoid the combinatorial blow-up of large libraries, while the latent-plus-multi-step part is meant to improve robustness when measurements are noisy or incomplete. That combination is the actual technical step forward. The abstract claims the method recovers accurate governing equations on nonlinear and chaotic systems and scales more favorably than standard SINDy. If the full experiments deliver clean quantitative comparisons with error bars and proper baselines, those two changes could be useful for people who already run SINDy pipelines on robotics or scientific data. The main soft spot is that the circuits only produce polynomials of bounded degree. Nothing in the construction adds sin, exp, or other transcendental terms that standard SINDy libraries include by default. If the test systems happen to be polynomial (Lorenz, Rössler, etc.), the comparison is not general; the paper would need to show either that the approximation error stays small or that the method still wins on systems with non-polynomial nonlinearities. The abstract also supplies no metrics, no ablation on the circuit depth or sparsity penalty, and no detail on how the graph sparsity is actually enforced, so the scaling claim rests on unshown evidence. This is the kind of paper that belongs in a reading group for people working on system identification methods. It is worth a serious referee if the experiments contain the missing quantitative results and address the polynomial limitation directly; otherwise it risks being a modest reformulation without clear gains.

Referee Report

2 major / 1 minor

Summary. The paper introduces AC-SINDy, a compositional extension of SINDy that replaces explicit feature libraries with arithmetic circuits built from compositions of linear functions and multiplicative interactions. This allows sparsity to be enforced directly over the computational graph. It further proposes separating state estimation from dynamics identification via latent state inference, shared dynamics, and multi-step supervision. Experiments on nonlinear and chaotic systems are reported to recover accurate, interpretable governing equations with more favorable scaling than standard SINDy.

Significance. If substantiated, the compositional circuit approach could address scalability limitations of library-based SINDy by avoiding explicit enumeration of basis functions while preserving interpretability through the graph structure. The latent-state formulation may offer improved noise robustness. However, the absence of quantitative support leaves the practical gains prospective rather than demonstrated.

major comments (2)

[Abstract] Abstract: the central claim that 'experiments on nonlinear and chaotic systems demonstrate that the method recovers accurate and interpretable governing equations while scaling more favorably than standard SINDy' is unsupported by any reported error metrics, error bars, baseline comparisons, or details on how sparsity is enforced over the circuit; the soundness assessment therefore rests on unshown evidence.
[Method] Method description (arithmetic circuit construction): the nodes consist of linear maps and multiplications, which generate only multivariate polynomials of bounded degree; the manuscript does not address how non-polynomial terms (e.g., sin, exp) that appear in standard SINDy libraries are represented or approximated, raising a risk that the claimed generality does not hold for systems whose nonlinearities lie outside this class.

minor comments (1)

[Abstract] Abstract: the description of the latent-state formulation could briefly note the supervision strategy (multi-step) to clarify how robustness is achieved.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments on our work. We provide point-by-point responses to the major comments below and indicate the revisions made to the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that 'experiments on nonlinear and chaotic systems demonstrate that the method recovers accurate and interpretable governing equations while scaling more favorably than standard SINDy' is unsupported by any reported error metrics, error bars, baseline comparisons, or details on how sparsity is enforced over the circuit; the soundness assessment therefore rests on unshown evidence.

Authors: We acknowledge the referee's concern regarding the support for the abstract's claim. Upon review, the experiments section does include demonstrations on nonlinear and chaotic systems with figures showing the recovered equations and scaling advantages. However, to ensure the claim is fully substantiated with quantitative evidence, we have updated the abstract to be more precise and have included additional error metrics, error bars, and explicit baseline comparisons in the revised manuscript. We have also elaborated on the sparsity enforcement over the arithmetic circuit in the methods section. revision: yes
Referee: [Method] Method description (arithmetic circuit construction): the nodes consist of linear maps and multiplications, which generate only multivariate polynomials of bounded degree; the manuscript does not address how non-polynomial terms (e.g., sin, exp) that appear in standard SINDy libraries are represented or approximated, raising a risk that the claimed generality does not hold for systems whose nonlinearities lie outside this class.

Authors: The referee is correct that the arithmetic circuit nodes, consisting of linear maps and multiplications, generate multivariate polynomials. This compositional approach enables scalable feature construction without explicit library enumeration while maintaining interpretability via the graph. Standard SINDy libraries often include non-polynomial terms, but for the systems considered in our experiments, polynomial representations suffice and yield accurate recoveries. We have added a paragraph in the discussion section addressing the method's applicability to polynomial nonlinearities and noting that non-polynomial terms can be approximated or handled by extending the circuit with additional operations in future extensions. revision: partial

Circularity Check

0 steps flagged

No significant circularity; method construction and evaluation are independent.

full rationale

The paper defines AC-SINDy by replacing SINDy libraries with arithmetic circuits composed of linear maps and multiplications, then applies latent-state inference and multi-step supervision. This construction is presented as a direct algorithmic extension without any fitted parameter being relabeled as a prediction, without self-citation chains supporting the core uniqueness or completeness claims, and without any equation that reduces the reported recovery accuracy to quantities already fitted on the same data. Experiments are described as separate validation on nonlinear/chaotic systems. No load-bearing step collapses by definition or by self-reference to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that arithmetic-circuit compositions suffice to span the needed function space and that latent-state inference can be decoupled from dynamics learning without introducing bias that affects equation recovery. No explicit free parameters or invented entities are named in the abstract.

axioms (2)

domain assumption Arithmetic circuits built from linear maps and multiplications can represent the nonlinear terms required for the target dynamical systems.
Invoked when the method replaces explicit libraries with circuit compositions.
domain assumption Latent-state inference combined with multi-step supervision yields dynamics estimates that remain interpretable and accurate under noise.
Stated as the basis for the robustness improvement.

pith-pipeline@v0.9.0 · 5404 in / 1343 out tokens · 26213 ms · 2026-05-10T04:35:02.451836+00:00 · methodology

discussion (0)

Reference graph

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