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Knowledge-Informed Kernel State Reconstruction for Interpretable Dynamical System Discovery

Luca Muscarnera, Silas Ruhrberg Estévez, Samuel Holt, Evgeny Saveliev, Mihaela van der Schaar

TL;DR

The paper addresses discovering dynamical systems from data when observations are sparse, noisy, and partially observed, modeling the latent state as $\dot{x}(t)=f(x(t))$ with measurements $y_i$ through $\mathcal{H}_i$. It introduces MAAT, a knowledge-informed kernel regression framework that embeds states in an RKHS and enforces physical priors to produce smooth trajectories and analytic derivatives for downstream symbolic regression. MAAT provides a calibrated surrogate for reconstruction error, yields robust derivative estimates, and improves identifiability of sparse symbolic forms across twelve benchmarks, outperforming baselines including SINDy, Kalman filters, GPs, Neural ODEs, PINNs, and PIKL. By injecting priors such as non-negativity and conservation laws, MAAT enhances interpretability and data efficiency in mechanistic modeling for domains like pharmacology, epidemiology, and physiology. This approach offers a principled path to interpretable, data-efficient discovery of governing equations in scientific and clinical contexts with fragmented measurements.

Abstract

Recovering governing equations from data is central to scientific discovery, yet existing methods often break down under noisy, partial observations, or rely on black-box latent dynamics that obscure mechanism. We introduce MAAT (Model Aware Approximation of Trajectories), a framework for symbolic discovery built on knowledge-informed Kernel State Reconstruction. MAAT formulates state reconstruction in a reproducing kernel Hilbert space and directly incorporates structural and semantic priors such as non-negativity, conservation laws, and domain-specific observation models into the reconstruction objective, while accommodating heterogeneous sampling and measurement granularity. This yields smooth, physically consistent state estimates with analytic time derivatives, providing a principled interface between fragmented sensor data and symbolic regression. Across twelve diverse scientific benchmarks and multiple noise regimes, MAAT substantially reduces state-estimation MSE for trajectories and derivatives used by downstream symbolic regression relative to strong baselines.

Knowledge-Informed Kernel State Reconstruction for Interpretable Dynamical System Discovery

TL;DR

The paper addresses discovering dynamical systems from data when observations are sparse, noisy, and partially observed, modeling the latent state as with measurements through . It introduces MAAT, a knowledge-informed kernel regression framework that embeds states in an RKHS and enforces physical priors to produce smooth trajectories and analytic derivatives for downstream symbolic regression. MAAT provides a calibrated surrogate for reconstruction error, yields robust derivative estimates, and improves identifiability of sparse symbolic forms across twelve benchmarks, outperforming baselines including SINDy, Kalman filters, GPs, Neural ODEs, PINNs, and PIKL. By injecting priors such as non-negativity and conservation laws, MAAT enhances interpretability and data efficiency in mechanistic modeling for domains like pharmacology, epidemiology, and physiology. This approach offers a principled path to interpretable, data-efficient discovery of governing equations in scientific and clinical contexts with fragmented measurements.

Abstract

Recovering governing equations from data is central to scientific discovery, yet existing methods often break down under noisy, partial observations, or rely on black-box latent dynamics that obscure mechanism. We introduce MAAT (Model Aware Approximation of Trajectories), a framework for symbolic discovery built on knowledge-informed Kernel State Reconstruction. MAAT formulates state reconstruction in a reproducing kernel Hilbert space and directly incorporates structural and semantic priors such as non-negativity, conservation laws, and domain-specific observation models into the reconstruction objective, while accommodating heterogeneous sampling and measurement granularity. This yields smooth, physically consistent state estimates with analytic time derivatives, providing a principled interface between fragmented sensor data and symbolic regression. Across twelve diverse scientific benchmarks and multiple noise regimes, MAAT substantially reduces state-estimation MSE for trajectories and derivatives used by downstream symbolic regression relative to strong baselines.
Paper Structure (61 sections, 6 theorems, 54 equations, 3 figures, 14 tables)

This paper contains 61 sections, 6 theorems, 54 equations, 3 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Dynamical System and Observation Model
  4. Reconstruction and Discovery Objectives
  5. Physical Priors
  6. Optimization Framework
  7. Knowledge-informed Kernel Regression
  8. Kernel State Reconstruction (KSR)
  9. Theoretical Justification
  10. RELATED WORK
  11. EXPERIMENTS
  12. Penalizing Derivative Magnitude
  13. Non-negativity Constraints
  14. Incorporating Dynamical System Structure Knowledge
  15. DISCUSSION
  16. ...and 46 more sections

Key Result

Lemma 1

Let $H:\mathbb{R}^d\!\to\!\mathbb{R}^p$ be a bounded linear observation operator. For any candidate trajectory $\hat{x}\in L^2([0,T];\mathbb{R}^d)$ and true trajectory $x$, define the risk $\mathcal{R}(\hat{x})=\|x-\hat{x}\|_{L^2}^2+\|H(x-\hat{x})\|_{L^2}^2$. Then Hence minimizing $\mathcal{R}$ is equivalent to minimizing the $L^2$ reconstruction error up to a constant factor.

Figures (3)

  • Figure 1: Overview of MAAT. Sparse anchor measurements and dense aggregate observations are combined with physical priors to produce reconstructed states through knowledge-informed kernel regression yielding smooth trajectories and analytical derivatives that can be used for symbolic regression.
  • Figure 2: State and Derivative Reconstruction on the EPO System under Noise. Ground truth (light blue), noisy observations (grey), and MAAT's reconstruction (dark blue). Even with significant noise, our method recovers a smooth, accurate trajectory.
  • Figure 3: Effect of noise on Lasso regression. Test error decays monotonically for low noise regimes while for high noise regimes the same behavior requires trespassing the interpolation threshold, leading to the need of a number of data points proportional to the number of features in the input space.

Theorems & Definitions (12)

  • Lemma 1: Composite loss is a calibrated surrogate
  • proof : Sketch
  • Proposition 1: FD noise floor vs KSR
  • Lemma 2: Generalized Representation Theorem
  • proof
  • Definition 1: Equivalence Between Distances
  • Corollary 1
  • proof
  • Lemma 3: Composite loss is a calibrated surrogate
  • proof
  • ...and 2 more