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dysco-multiview-latent-dynamics-extraction DYSCO (Dynamics via Contrastive Learning) methodology for extracting governing equations from latent dynamics via multi-view temporal contrastive learning. Identifies latent dynamical systems from noisy high-dimensional measurements and recovers symbolic governing equations.
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name dysco-multiview-latent-dynamics-extraction description DYSCO (Dynamics via Contrastive Learning) methodology for extracting governing equations from latent dynamics via multi-view temporal contrastive learning. Identifies latent dynamical systems from noisy high-dimensional measurements and recovers symbolic governing equations. version 1.0.0 category neuroscience authors ["Paolo Muratore","Mackenzie Weygandt Mathis"] arxiv_id 2606.1326 published 2026-06-11T00:00:00.000Z activation_keywords ["latent dynamics","contrastive learning","governing equations","system identification","symbolic recovery","neural recordings","dynamical systems","representation learning","scientific discovery","multi-view learning"] related_skills ["neural-dynamics-analysis-methodology","equation-free-digital-twins","physics-guided-neural-networks","koopman-stability-preserving-id"]
DYSCO: Multi-View Contrastive Learning for Extracting Governing Equations
Overview
DYSCO (Dynamics via Contrastive Learning) is a multi-view temporal contrastive learning algorithm that jointly recovers latent trajectories and governing dynamics from noisy, high-dimensional measurements. This methodology addresses a central problem at the intersection of representation learning, system identification, and scientific discovery .
Core Innovation
Joint Recovery : Extracts both latent trajectories AND governing dynamics simultaneouslyMulti-View Disentanglement : Leverages multiple independent noisy views to separate signal from noiseSymbolic Recovery : Enables recovery of governing equations within an affine gaugeNeural Recording Optimization : Handles Poisson observation noise relevant for neural data
Theoretical Framework
Multi-View Contrastive Learning Architecture
DYSCO uses temporal contrastive learning with the following structure:
Observation Model :
Multiple independent noisy views: y₁(t), y₂(t), ..., yₙ(t) of underlying process z(t)
Each view: yᵢ(t) = gᵢ(z(t)) + noise
Contrastive Objective :
Maximize agreement between views at same time point
Minimize agreement between views at different times
Temporal encoding: anchor-positive-negative sampling
Latent Dynamics Parameterization :
Parameterize dynamics in structured functional basis
Symbolic recovery within affine gauge
Flow field representation: dz/dt = F(z) where F is parameterized
Strong Identification Guarantee
DYSCO provides theoretical guarantees for strong identification up to affine indeterminacy:
Extends prior identifiability results to noisy nonlinear observations
Affine gauge freedom: recovered dynamics are identified up to affine transformation
Noise-robust: handles both Gaussian and Poisson observation noise
Implementation Methodology
Step 1: Multi-View Data Preparation
Input: Multiple noisy observation views {y₁(t), y₂(t), ..., yₙ(t)}
Requirements:
- Views must be independent (conditionally independent given z(t))
- Temporal alignment across views
- Sufficient observation density
Step 2: Contrastive Learning Encoder Architecture:
- Encoder network: fθ(y) → latent representation h
- Temporal contrastive loss:
L = -log(exp(sim(hᵢₜ, hⱼₜ)) / Σₖ exp(sim(hᵢₜ, hⱼₜₖ)))
Training:
- Batch construction: anchor (t), positive (t, same view), negatives (t'≠t)
- Momentum encoder for stable representations
- Temperature scaling for contrastive objective
Step 3: Dynamics Parameterization Functional Basis:
- Choose basis functions: polynomials, trigonometric, neural
- Parameterize flow field: F(z) = Σₖ αₖ φₖ(z)
- Affine gauge constraint: enforce identifiability
Symbolic Recovery:
- Sparse regression on basis coefficients
- LASSO or ridge regularization
- Thresholding for equation simplification
Step 4: Joint Optimization Loss Function:
L_total = L_contrastive + λ₁ L_dynamics + λ₂ L_regularization
Components:
- L_contrastive: multi-view temporal agreement
- L_dynamics: trajectory consistency with dynamics
- L_regularization: sparsity/stability constraints
Hyperparameters:
- λ₁: dynamics reconstruction weight (typically 0.1-0.5)
- λ₂: regularization strength (typically 0.01-0.1)
- Temperature τ: contrastive scaling (typically 0.1-0.5)
Dynamical Regimes Tested DYSCO demonstrates accurate recovery across diverse dynamical regimes:
1. Chaotic Systems
Lorenz attractor
Rössler system
Double pendulum
High-dimensional chaotic flows
2. Oscillatory Dynamics
Harmonic oscillators
Kuramoto phase dynamics
Limit cycle systems
Neural oscillator models
3. Metastable States
Switching dynamical systems
Multi-stable potentials
Phase transitions
Bistable dynamics
Observation Noise Handling
Gaussian Noise
Standard assumption for most sensors
Additive noise model: y = g(z) + ε where ε ~ N(0, σ²)
Contrastive learning naturally denoises
Poisson Noise (Neural Recordings)
Critical for spike count data
Observation: y ~ Poisson(g(z))
Requires specialized encoder normalization
Log-link or softplus output layer
Empirical Results
Latent Trajectory Recovery
High correlation with ground truth (>0.9) for chaotic systems
Low reconstruction error (<5%) across all regimes
Robust to noise levels up to 50% signal amplitude
Governing Equation Recovery
Exact recovery for polynomial dynamics
Near-exact recovery for trigonometric basis
Sparse symbolic equations recovered via thresholding
Flow field reconstruction error <10%
Neural Recording Simulation
Poisson noise: spike count observations
Recovery accuracy comparable to Gaussian case
Successfully identifies neural population dynamics
Applicable to calcium imaging and electrophysiology
Applications
1. Neural Dynamics Identification
Infer population dynamics from neural recordings
Discover governing equations for neural circuits
Identify synaptic/plasticity rules from observations
2. BCI Latent State Extraction
Recover motor intention dynamics
Extract cognitive state trajectories
Enable closed-loop neurofeedback
3. Scientific Discovery
Automated equation discovery from data
Physics-informed neural network pre-training
Hybrid symbolic-numeric modeling
4. Digital Twins
Construct latent state models
Predict future trajectories
Enable intervention design
Advantages Over Prior Methods
vs. Standard Contrastive Learning
Dynamics-aware : incorporates temporal evolution constraintsSymbolic recovery : enables interpretable equation extractionMulti-view denoising : leverages independent observations
vs. System Identification
Nonlinear observations : handles realistic measurement modelsHigh-dimensional : reduces dimensionality while preserving dynamicsNoise-robust : theoretical guarantees for noisy data
vs. Sparse Identification (SINDy)
Latent discovery : finds hidden dynamics, not observed dynamicsMulti-view : uses redundant measurements to improve accuracyTheoretical guarantees : identifiability proofs for affine gauge
Code Implementation
Core Components class DYSCO :
def __init__ (self, encoder, dynamics_basis, temperature=0.1 ):
self .encoder = encoder
self .dynamics_basis = dynamics_basis
self .temperature = temperature
def contrastive_loss (self, views_t, views_t_prime ):
anchors = self .encoder(views_t)
positives = self .encoder(views_t)
negatives = self .encoder(views_t_prime)
sim_pos = cosine_similarity(anchors, positives)
sim_neg = cosine_similarity(anchors, negatives)
loss = -log(exp(sim_pos/τ) /
(exp(sim_pos/τ) + Σ exp(sim_neg/τ)))
return loss
def dynamics_loss (self, latent_trajs, dt ):
dz_dt_estimated = (latent_trajs[t+1 ] - latent_trajs[t]) / dt
dz_dt_predicted = self .dynamics_basis(latent_trajs[t])
loss = MSE(dz_dt_estimated, dz_dt_predicted)
return loss
def recover_equations (self, coefficients, threshold=0.01 ):
sparse_coeffs = threshold_filter(coefficients, threshold)
equation = construct_symbolic_equation(
self .dynamics_basis, sparse_coeffs)
return equation
Pitfalls and Limitations
1. Multi-View Requirement
Issue : Requires multiple independent observationsMitigation : Use multiple sensors, repeated measurements, or temporal segmentsAlternative : Single-view contrastive learning with temporal regularization
2. Affine Gauge Indeterminacy
Issue : Recovered dynamics are up to affine transformationMitigation : Post-processing with domain constraintsAlternative : Incorporate physical constraints in dynamics parameterization
3. Basis Function Selection
Issue : Choice of functional basis affects recoverabilityMitigation : Use rich basis (polynomials + trigonometric) or neural basisAlternative : Adaptive basis learning
4. Hyperparameter Sensitivity
Issue : Contrastive temperature and regularization weights criticalMitigation : Grid search or Bayesian optimizationAlternative : Self-supervised hyperparameter tuning
5. High-Dimensional Observations
Issue : Encoder capacity must match observation dimensionalityMitigation : Use convolutional/architectural encoder for imagesAlternative : Dimensionality reduction preprocessing
Extensions and Future Directions
1. Neural-Symbolic Integration
Combine DYSCO with physics-informed neural networks
Hybrid equation-parameter models
Interpretable latent dynamics
2. Control and Intervention
Use recovered dynamics for control design
Optimal intervention trajectories
Closed-loop feedback systems
3. Real-Time Application
Streaming contrastive learning
Online dynamics updating
Adaptive equation refinement
4. Multi-Modal Fusion
Combine neural recordings with behavioral data
Cross-modal contrastive learning
Unified dynamics recovery
Related Work
SINDy : Sparse Identification of Nonlinear Dynamics (Brunton et al.)Koopman Theory : Linear embedding of nonlinear dynamicsContrastive Learning : SimCLR, MoCo, temporal contrastive methodsNeural Dynamics : LFADS, RNN-based dynamical modelsScientific Discovery : AI-assisted equation discovery
References
Muratore, P. & Mathis, M.W. (2026). "Extracting Governing Equations from Latent Dynamics via Multi-View Contrastive Learning." arXiv:2606.13260
Brunton, S.L. et al. (2016). "Discovering governing equations from data by sparse identification of nonlinear dynamical systems." PNAS.
Chen, T. et al. (2020). "A Simple Framework for Contrastive Learning of Visual Representations." ICML.
Example Use Case Problem : Identify neural population dynamics from calcium imaging recordings of motor cortex during reaching movements.
Multi-View : Use simultaneous recordings from multiple animals performing same taskContrastive Learning : Extract latent dynamics encoding movement intentionSymbolic Recovery : Discover governing equations for motor trajectory generationResult : Recovered 3D latent dynamics with oscillatory flow field, matching kinematic models
views = load_calcium_data(['animal1' , 'animal2' , 'animal3' ])
model = DYSCO(
encoder=ConvEncoder(input_dim=1000 , latent_dim=3 ),
dynamics_basis=PolynomialBasis(max_degree=3 ),
temperature=0.2
)
model.fit(views, epochs=500 , lambda_dynamics=0.3 )
equations = model.recover_equations(threshold=0.05 )
