Dysco Multiview Latent Dynamics Extraction

---
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: 2606.13260
published: 2026-06-11
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 simultaneously
- **Multi-View Disentanglement**: Leverages multiple independent noisy views to separate signal from noise
- **Symbolic Recovery**: Enables recovery of governing equations within an affine gauge
- **Neural 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:

1. **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`

2. **Contrastive Objective**:
   - Maximize agreement between views at same time point
   - Minimize agreement between views at different times
   - Temporal encoding: anchor-positive-negative sampling

3. **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 constraints
- **Symbolic recovery**: enables interpretable equation extraction
- **Multi-view denoising**: leverages independent observations

### vs. System Identification
- **Nonlinear observations**: handles realistic measurement models
- **High-dimensional**: reduces dimensionality while preserving dynamics
- **Noise-robust**: theoretical guarantees for noisy data

### vs. Sparse Identification (SINDy)
- **Latent discovery**: finds hidden dynamics, not observed dynamics
- **Multi-view**: uses redundant measurements to improve accuracy
- **Theoretical guarantees**: identifiability proofs for affine gauge

## Code Implementation

### Core Components
```python
class DYSCO:
    def __init__(self, encoder, dynamics_basis, temperature=0.1):
        self.encoder = encoder  # fθ: y → h
        self.dynamics_basis = dynamics_basis  # φₖ(z)
        self.temperature = temperature
        
    def contrastive_loss(self, views_t, views_t_prime):
        # Multi-view temporal contrastive objective
        anchors = self.encoder(views_t)
        positives = self.encoder(views_t)  # same time
        negatives = self.encoder(views_t_prime)  # different time
        
        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):
        # Flow field consistency
        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):
        # Symbolic equation extraction
        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 observations
- **Mitigation**: Use multiple sensors, repeated measurements, or temporal segments
- **Alternative**: Single-view contrastive learning with temporal regularization

### 2. Affine Gauge Indeterminacy
- **Issue**: Recovered dynamics are up to affine transformation
- **Mitigation**: Post-processing with domain constraints
- **Alternative**: Incorporate physical constraints in dynamics parameterization

### 3. Basis Function Selection
- **Issue**: Choice of functional basis affects recoverability
- **Mitigation**: Use rich basis (polynomials + trigonometric) or neural basis
- **Alternative**: Adaptive basis learning

### 4. Hyperparameter Sensitivity
- **Issue**: Contrastive temperature and regularization weights critical
- **Mitigation**: Grid search or Bayesian optimization
- **Alternative**: Self-supervised hyperparameter tuning

### 5. High-Dimensional Observations
- **Issue**: Encoder capacity must match observation dimensionality
- **Mitigation**: Use convolutional/architectural encoder for images
- **Alternative**: 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 dynamics
- **Contrastive Learning**: SimCLR, MoCo, temporal contrastive methods
- **Neural Dynamics**: LFADS, RNN-based dynamical models
- **Scientific 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.

**DYSCO Application**:
1. **Multi-View**: Use simultaneous recordings from multiple animals performing same task
2. **Contrastive Learning**: Extract latent dynamics encoding movement intention
3. **Symbolic Recovery**: Discover governing equations for motor trajectory generation
4. **Result**: Recovered 3D latent dynamics with oscillatory flow field, matching kinematic models

**Implementation**:
```python
# Load calcium imaging from multiple animals
views = load_calcium_data(['animal1', 'animal2', 'animal3'])

# Initialize DYSCO
model = DYSCO(
    encoder=ConvEncoder(input_dim=1000, latent_dim=3),
    dynamics_basis=PolynomialBasis(max_degree=3),
    temperature=0.2
)

# Train
model.fit(views, epochs=500, lambda_dynamics=0.3)

# Recover equations
equations = model.recover_equations(threshold=0.05)
# Output: dz₁/dt = α₁z₂ + α₂z₁z₃
#         dz₂/dt = α₃z₁ - α₄z₂²
#         dz₃/dt = α₅sin(z₁) + α₆z₃
```