Extracting Governing Equations from Latent Dynamics via Multi-View Contrastive Learning
Paolo Muratore, Mackenzie Weygandt Mathis
Abstract
Identifying latent dynamical systems from noisy, high-dimensional measurements is a central problem at the intersection of representation learning, system identification, and scientific discovery. We present DYSCO, a multi-view temporal contrastive learning algorithm that jointly recovers latent trajectories and the governing dynamics from such observations, by leveraging multiple independent noisy views of the same underlying process to disentangle signal from noise. By parameterizing the dynamics in a structured functional basis, our framework further enables symbolic recovery of the governing equations within an affine gauge. We offer theoretical guarantees for strong identification up to an affine indeterminacy, extending prior identifiability results to the realistic setting of noisy nonlinear observations. Empirically, we demonstrate accurate recovery of both latent trajectories and flow fields across a diverse set of dynamical regimes (e.g., chaotic, oscillatory, and metastable) under both Gaussian and Poisson observation noise, the latter being particularly relevant for neural recordings.
AI Impact Assessments
(1 models)Scientific Impact Assessment: DYSCO — Extracting Governing Equations from Latent Dynamics via Multi-View Contrastive Learning
1. Core Contribution
DYSCO addresses a fundamental inverse problem: jointly recovering latent states and their governing dynamical equations from noisy, high-dimensional observations. The key novelty is the combination of three elements into a unified framework: (i) multi-view temporal contrastive learning for denoising through cross-view consistency, (ii) structured symbolic parameterization of dynamics via a functional basis (polynomial library à la SINDy), and (iii) theoretical identifiability guarantees up to an affine transformation. The central insight is that multiple independent noisy views of the same latent process provide a principled mechanism to separate signal from noise — something prior contrastive identification work (DYNCL, Laiz et al. 2025) could not handle since it assumed noiseless observations. This is a meaningful extension because real-world measurements are invariably noisy.
2. Methodological Rigor
Theoretical Analysis: Theorem 1 establishes that in the asymptotic limit (V→∞, T→∞), the latent state and dynamics are identifiable up to an affine transformation. The proof is well-structured across five steps: (1) view-averaging denoises observations via sub-Gaussian concentration, (2) the posterior concentrates on the true latent state through bi-Lipschitz inversion, (3) joint realizability reduces multi-horizon to single-step optimization, (4) the Bayes-optimal score converges to the clean-observation score, and (5) score matching forces affine recovery via the Jacobian constancy argument from Laiz et al. (2025).
However, there are notable gaps between theory and practice. The theorem requires V→∞, yet experiments use V=2 views. The paper acknowledges this through "amortized denoising" — a heuristic rather than a formal guarantee. Assumptions A1 (joint realizability) and A2 (stable C¹ limit) are strong and difficult to verify in practice. The theoretical framework also assumes Gaussian latent noise and sub-Gaussian observation noise, while experiments include Poisson noise that violates these assumptions.
Experimental Validation: The experiments cover seven dynamical systems spanning chaotic, oscillatory, and metastable regimes — a reasonably comprehensive testbed. The evaluation protocol is appropriate: affine-aligned R² and dynR² metrics respect the theoretical gauge freedom. Ablation studies on number of views, noise intensity, and integration horizon are informative. The comparison against DYNCL on unforced systems demonstrates clear advantages under noise, though the comparison is limited to only two systems (the others use forcing, which DYNCL doesn't support).
A weakness is that all experiments are synthetic. The mixing function g is a randomly initialized MLP, which may not represent the complexity of real measurement processes. The paper lacks comparison with other relevant baselines (e.g., Champion et al. 2019's VAE+SINDy approach, LFADS, or other sequential methods).
3. Potential Impact
Neuroscience Applications: The framework is positioned for neural data analysis, where trial-structured recordings naturally provide multiple views of shared latent dynamics. The Poisson noise model (neural spiking) is practically relevant. If validated on real neural data, this could provide a principled path from raw recordings to interpretable dynamical models — directly addressing the gap between implementation-level and algorithmic-level descriptions in Marr's framework.
Broader Scientific Discovery: The symbolic recovery capability, while currently proof-of-concept quality (Appendix C shows imperfect recovery with spurious terms), opens an interesting direction for automated equation discovery from indirect measurements. This extends the SINDy paradigm to latent, noisy settings with identifiability guarantees.
Methodological Influence: The multi-view denoising mechanism could influence broader self-supervised learning practices. The theoretical framework connecting multi-view consistency to identifiability under noise extends the nonlinear ICA literature in a meaningful direction.
4. Timeliness & Relevance
This paper addresses a genuine bottleneck: existing symbolic regression methods (SINDy and variants) typically require direct state access, while existing latent dynamics methods (LFADS, CEBRA) don't recover explicit equations. The intersection — symbolic identification from noisy latent observations with guarantees — is relatively unexplored. The work is timely given growing interest in interpretable AI, neural population dynamics, and the identifiability of learned representations. The connection to emerging contrastive learning theory (Zimmermann et al. 2021, Laiz et al. 2025) places it well within current research momentum.
5. Strengths & Limitations
Strengths:
Limitations:
Notable Observations:
Overall Assessment
DYSCO makes a solid theoretical and methodological contribution by extending contrastive identification to noisy observations through multi-view consistency. The theoretical framework is elegant, and the empirical results on synthetic data are convincing for trajectory recovery. However, the symbolic recovery — arguably the most novel and impactful claimed contribution — remains underdeveloped. The paper's impact would be substantially strengthened by real-data validation and more robust symbolic regression within the affine gauge. As it stands, this is a meaningful incremental advance over DYNCL with a promising but incompletely realized vision.
Generated Jun 12, 2026
Comparison History (26)
Paper 2 has higher potential impact due to a broader, more foundational contribution: it extends learning theory beyond (approximate) independence via simulatable processes, recovers VC-style guarantees, and links regret to time-bounded Kolmogorov complexity—ideas likely to influence multiple areas (PAC learning, online learning, conditional sampling, computational learning theory). Its applicability spans any domain with dependent/complex data where simulators exist, making it timely and broadly relevant. Paper 1 is strong and useful for scientific discovery in dynamical systems, but is more specialized and likely narrower in cross-field impact.
Paper 2 directly tackles a fundamental problem in scientific discovery: extracting governing equations from noisy, high-dimensional data. Its approach bridges machine learning with physics, biology, and neuroscience, offering broad, cross-disciplinary applications. While Paper 1 presents strong algorithmic improvements for sequence generation, Paper 2's potential to uncover fundamental laws of nature from observational data provides a more profound potential impact across diverse scientific fields.
Paper 2 likely has higher impact: it introduces a broadly applicable framework (multi-view temporal contrastive learning) that connects representation learning with system identification and scientific equation discovery, with theoretical identifiability guarantees under noisy nonlinear observations and demonstrated performance across diverse dynamical regimes and noise models (including Poisson, relevant to neuroscience). This can influence multiple fields (ML, dynamical systems, physics-informed learning, neuroscience). Paper 1 is rigorous and valuable for distributed/federated optimization, but is more specialized to ASGD robustness and primarily impacts optimization systems literature.
Paper 1 addresses a fundamental and timely problem in LLM post-training by bridging interpretability and alignment, offering a practical pipeline that diagnoses spurious correlations, mitigates sycophancy, and shapes model behavior at the concept level. Given the enormous current investment in LLM alignment and the broad applicability across the AI industry, this work has potential for very wide adoption. Paper 2 is methodologically strong with nice theoretical guarantees for latent dynamics recovery, but targets a narrower scientific audience. Paper 1's combination of novelty (unifying interpretability with training signal design), practical utility, and timeliness in the rapidly growing alignment field gives it higher expected impact.
Paper 1 has higher potential impact due to its broad applicability across scientific disciplines, such as physics and neuroscience, aligning strongly with the transformative 'AI for Science' paradigm. By extracting symbolic governing equations from noisy, high-dimensional data with theoretical guarantees, it addresses a fundamental problem in scientific discovery. While Paper 2 provides valuable theoretical insights into Graph Neural Network expressivity, its contributions are largely confined to the graph machine learning community, making it less interdisciplinary and transformative than Paper 1.
Paper 2 has higher impact potential: it tackles a broadly important and timely problem (recovering latent dynamics and governing equations from noisy high-dimensional data) with clear real-world applications across physics, neuroscience, and engineering. The multi-view contrastive framework plus structured basis enabling symbolic equation recovery is both novel and practically useful, and it offers identifiability guarantees under realistic noise and nonlinear observations. Its breadth and applicability likely exceed Paper 1’s more specialized gauge-invariant neural layer for cup-product observables, whose impact may be narrower despite strong theoretical rigor.
DYSCO addresses a fundamental problem in scientific discovery—extracting governing equations from noisy high-dimensional data—with theoretical identifiability guarantees and broad applicability across scientific domains (neuroscience, physics, biology). Its contributions span representation learning, system identification, and symbolic regression, offering lasting methodological impact. Paper 2, while impressive in achieving gold-medal-level math competition performance, is more narrowly focused on benchmark achievement through engineering (test-time scaling, tournament selection) rather than introducing fundamentally new scientific principles. Paper 1's theoretical framework and cross-disciplinary relevance give it broader long-term impact.
Paper 2 (DYSCO) addresses a fundamental problem at the intersection of multiple fields—representation learning, system identification, and scientific discovery—with both theoretical guarantees and broad empirical validation. Its ability to recover governing equations from noisy observations has wide-ranging applications in physics, neuroscience, and other sciences. Paper 1 provides useful mechanistic insights into RL post-training for reasoning models, but is more narrowly focused on understanding existing training practices for a specific model. Paper 2's theoretical contributions (identifiability guarantees) and methodological novelty (multi-view contrastive learning for dynamics) give it broader and more lasting impact.
Paper 2 addresses a fundamental challenge in AI for Science by enabling the discovery of governing equations from noisy, high-dimensional data. Its theoretical guarantees for identifiability and broad applicability across diverse fields like neuroscience and physics give it a foundational scientific impact. While Paper 1 is highly practical and timely for LLM deployment, it is primarily a systems engineering optimization with a narrower scope of impact.
While Paper 1 offers impressive scalability and speedups for practical forecasting, Paper 2 tackles a fundamental bottleneck in scientific discovery: extracting symbolic governing equations from high-dimensional, noisy data. By bridging representation learning with symbolic regression and providing theoretical identifiability guarantees, Paper 2 has a profound potential impact across physics, neuroscience, and biology, making it more transformative for fundamental science.