Source-linked AI summary
Discovering State Variables Hidden in Experimental Data
Boyuan Chen, Kuang Huang, Sunand Raghupathi, Ishaan Chandratreya, Qiang Du, Hod Lipson
TL;DR
The paper tackles the problem of identifying hidden state variables directly from high-dimensional video, without assuming the relevant variables beforehand. It estimates intrinsic dimension and uses that estimate to identify Neural State Variables, demonstrating accurate dynamics capture and stable long-term prediction across physical systems. The approach nevertheless has prediction-quality limitations in some settings, including rapid degradation during rollouts.
Problem
Identifying complete and non-redundant state variables from raw observations remains difficult because many data-driven methods assume the relevant variables are already provided.
Method
A two-stage video-based approach first estimates intrinsic dimension, then learns Neural State Variables with a latent reconstruction network of the estimated dimension.
Results
The method discovers candidate state variables that accurately capture system dynamics and provide stable long-term predictions, including across varied physical dynamical systems.
Takeaways & Limitations
Neural State Variables provide a compact representation for modeling dynamics and evaluating stable long-term predictions from visual observations.
Takeaways & Limitations
Prediction quality can deteriorate rapidly during rollouts, limiting the approach in settings where outputs become degraded after only a few steps.
Abstract
from arXiv · showhide
All physical laws are described as relationships between state variables that give a complete and non-redundant description of the relevant system dynamics. However, despite the prevalence of computing power and AI, the process of identifying the hidden state variables themselves has resisted automation. Most data-driven methods for modeling physical phenomena still assume that observed data streams already correspond to relevant state variables. A key challenge is to identify the possible sets of state variables from scratch, given only high-dimensional observational data. Here we propose a new principle for determining how many state variables an observed system is likely to have, and what these variables might be, directly from video streams. We demonstrate the effectiveness of this approach using video recordings of a variety of physical dynamical systems, ranging from elastic double pendulums to fire flames. Without any prior knowledge of the underlying physics, our algorithm discovers the intrinsic dimension of the observed dynamics and identifies candidate sets of state variables. We suggest that this approach could help catalyze the understanding, prediction and control of increasingly complex systems. Project website is at: https://www.cs.columbia.edu/~bchen/neural-state-variables
Main Text
The paper addresses the difficulty of identifying hidden, complete, and non-redundant state variables from raw observations, rather than assuming them in advance. It proposes a video-based two-stage approach that estimates intrinsic dimensionality and identifies Neural State Variables, which capture dynamics and support stable long-term prediction.
- Main Text: The central challenge is that relevant state variables are hidden, non-unique, and difficult to identify for unfamiliar or complex systems from observational data.Existing approaches often assume measurements of the relevant state variables, while raw video contains many variables that are neither compact nor complete.
- Main Text: The method first estimates a system’s intrinsic dimension from video and then reconstructs a latent representation with exactly that dimension to identify Neural State Variables.The initial dynamics model uses bottleneck latent embeddings, and the second-stage reconstruction network uses the estimated intrinsic dimension.
- Main Text: Neural State Variables accurately capture overall system dynamics and can serve as an intermediate representation and evaluation metric for stable long-term prediction.The approach was evaluated quantitatively and qualitatively across nine physical dynamical systems from varied experimental domains.
- Main Text: The reduced-dimensional representation supports more stable long-term predictions than higher-dimensional autoencoders, which can produce blurred or plain-background frames after only a few rollout steps.The paper also presents a hybrid prediction scheme intended to combine accurate and stable long-term predictions.
- Main Text: The learned representation remains robust under various visual perturbations, extending its use beyond clean video observations.The paper reports this robustness as an additional property of Neural State Variables for modeling system dynamics.
Modeling Dynamical Systems from Videos
The paper models dynamical systems directly from video by learning compact latent representations that predict future frames, then uses intrinsic dimensionality to target the number of state variables. Video-based predictions are accurate across diverse physical systems and substantially outperform simple baselines.
- Modeling Dynamical Systems from Videos: Video observations avoid requiring prior knowledge of internal processes and can provide rich physics signals without specialized equipment.The authors suggest the approach could potentially extend to other sensor arrays.
- Modeling Dynamical Systems from Videos: The framework learns a compact latent space from high-dimensional video by encoding current frames and decoding predicted future frames.Accurate future prediction requires the bottleneck to retain sufficient information about system dynamics.
- Modeling Dynamical Systems from Videos: A two-stage strategy estimates intrinsic dimension from predictive latent vectors, then constrains a second encoder-decoder to discover Neural State Variables with that dimension.The identified variables are subsequently used to model dynamics and assess prediction stability.
- Modeling Dynamical Systems from Videos: The latent dimension is critical: oversized embeddings can overfit, limit long-range prediction, and obscure compact state variables, whereas undersized embeddings can under-fit.The method therefore aims to approach the exact number of state variables rather than selecting latent size by guesswork.
- Modeling Dynamical Systems from Videos: Across nine systems spanning pendulums, waves, multiphase flow, aeroelasticity, and combustion, predictions are accurate and substantially outperform linear extrapolation and input-copying baselines.For systems with known physical quantities, the model also predicts those variables accurately against ground truth.
Intrinsic Dimension Estimation
The method estimates intrinsic dimension from latent vectors because unknown systems do not provide state variables or dimensions in advance. Geometric manifold learning avoids direct latent shrinking and yields accurate estimates for known systems with intuitive relative rankings elsewhere.
- Intrinsic Dimension Estimation: Intrinsic dimension is the minimum number of independent variables needed to fully describe a dynamical system, equivalently the topological dimension of its state manifold.The paper treats prior knowledge of this dimension and its corresponding variables as unavailable for unknown or partially known systems.
- Intrinsic Dimension Estimation: Directly shrinking a predictive auto-encoder’s latent space fails because predictions deteriorate before the minimal state-variable set is reached.In the reported experiment, predictions broke down when the latent dimension was reduced directly to the intrinsic dimension.
- Intrinsic Dimension Estimation: The proposed alternative applies geometric manifold learning to predictive latent vectors, using Levina-Bickel’s algorithm to estimate intrinsic dimension after one network-training step.The estimator uses local neighbor-distance geometry on latent vectors collected from the trained dynamics model.
- Intrinsic Dimension Estimation: The method accurately estimates intrinsic dimension for all known systems and produces a reasonable, intuitive relative ranking for systems without ground-truth dimensions.The evaluation compares estimates with raw-image observations and partial ground truths.
- Intrinsic Dimension Estimation: Among compared intrinsic-dimensionality estimators, Levina-Bickel’s algorithm provides the most robust and reliable estimation.The comparison includes MiND ML, MiND KL, Hein, and CD implementations.
Neural State Variables
The paper introduces Neural State Variables, a two-stage approach that estimates a system’s intrinsic dimension and identifies compact variables from video. These variables capture system dynamics and support stable, accurate long-term prediction, including a hybrid scheme that balances stability and one-step accuracy.
- Prediction: The model produces accurate video predictions and substantially outperforms copying input data and linear extrapolation on physics evaluations.The elastic double pendulum evaluation uses 60-fps video and compares predicted-frame physical variables with ground truth.
- Neural State Variables: The two-stage framework estimates intrinsic dimension, then identifies a corresponding compact set of Neural State Variables from video dynamics.The first stage uses a dynamics-predictive model and geometric manifold learning; the second trains a latent reconstruction network at the estimated dimension.
- Neural State Variables: Neural State Variables capture sufficient information about complete system dynamics while using the intrinsic dimension, such as 4 for rigid and 6 for elastic double pendulums.The framework predicts accurate future frames from these super-compact variables.
- Stable Long-Term Prediction: Neural State Variables provide the most stable long-term predictions, but can have inferior one-step prediction accuracy than higher-dimensional latent embeddings.The hybrid scheme uses Neural State Variables as stabilizers and higher-dimensional embeddings for prediction, combining stable and accurate long-term behavior.
- Stability Indicators: The latent dynamics error correlates strongly with the physics reject ratio, providing an alternative video-based metric for long-term prediction stability without prior system knowledge.This metric evaluates how predicted dynamics deviate from the true state space over time.
- Robust Long-Term Prediction: Under visual perturbations, Neural State Variable rollouts remain the most robust across perturbation types and levels, whereas high-dimensional rollouts quickly become unstable.Lower-dimensional representations above the intrinsic dimension can sometimes recover stability, but require more prediction steps.
Analysis
The authors test whether Neural State Variables encode physical meaning by regressing conventional variables and visualizing latent structure. They report richer dynamical information than PCA-derived state variables and observe physical variables and symmetries in the learned representation.
- Using 30% of labeled data, Neural State Variables accurately regress conventional physical variables including positions, velocities, and energies.
- The comparison uses the same number of state variables, equal to the intrinsic dimension, and the same labeled data for both representations.
- PCA-derived state variables have much larger regression errors, especially for velocity variables, and can hardly capture system dynamics.
- Visualizations colored by physical-variable values show that the learned Neural State Variables capture physical variables and reveal inherent symmetrical structures.
Outlook
The outlook positions Neural State Variables as a step toward machine-assisted discovery of complete, non-redundant variables from raw visual observations without prior physical knowledge. It highlights stable long-term prediction and possible extensions for physical regularization, imperfect observations, interpretability, and control.
- The framework distills complete and non-redundant Neural State Variables from raw visual observations without assuming which physical variables or mathematical primitives to use.
- Neural State Variables support stable long-term predictions despite accumulated model prediction errors and external visual effects.
- Future work could regularize Neural State Variables with physical knowledge so they better correspond to traditional variables and satisfy constraints such as energy conservation (38–45).
- Corrupted or incomplete frames, hidden factors, system uncertainty, and inappropriate sampling frequency can prevent observations from fully capturing the real physics.
- Regularization methods may address imperfect observations, while further analysis could translate Neural State Variables into interpretable physics.
- Stable long-term prediction and learned state variables could serve as components of automated control systems.