Diffusion Models¶

This section covers thermodynamic diffusion models, which leverage principles of thermal diffusion and stochastic processes to generate samples and solve inverse problems.

Overview¶

Thermodynamic diffusion models extend traditional diffusion models by incorporating explicit thermodynamic state variables and physical constraints. These models can generate samples that evolve according to realistic physical processes while maintaining thermodynamic consistency.

Theoretical Foundation¶

Forward Diffusion Process¶

The forward process gradually adds noise according to a diffusion schedule: \(\(q(\mathbf{x}_t | \mathbf{x}_{t-1}) = \mathcal{N}(\mathbf{x}_t; \sqrt{1-\beta_t}\mathbf{x}_{t-1}, \beta_t \mathbf{I})\)\)

Where \(\beta_t\) is the noise schedule.

Thermodynamic Interpretation¶

In thermodynamic terms:

Energy: \(U_t = \|\mathbf{x}_t\|^2 / 2\)
Temperature: \(T_t = \beta_t / 2\)
Entropy: \(S_t = \frac{d}{2}\log(2\pi e T_t)\)
Free Energy: \(F_t = U_t - T_t S_t\)

Score Function¶

The score function represents the gradient of log-density: \(\(s_\theta(\mathbf{x}_t, t) = \nabla_{\mathbf{x}_t} \log p_t(\mathbf{x}_t)\)\)

Thermodynamic Score Models¶

Energy-Based Score¶

Define score in terms of energy: \(\(s_\theta(\mathbf{x}, t) = -\frac{1}{T_t}\nabla_{\mathbf{x}} U_\theta(\mathbf{x}, t)\)\)

Where \(U_\theta\) is a learned energy function.

Temperature-Dependent Score¶

Score function with explicit temperature dependence: \(\(s_\theta(\mathbf{x}, t) = -\frac{1}{T_t}\nabla_{\mathbf{x}} U_\theta(\mathbf{x}, t) + \sqrt{\frac{2}{T_t}}\boldsymbol{\xi}\)\)

Where \(\boldsymbol{\xi}\) represents thermal fluctuations.

Implementation¶

class ThermodynamicScoreModel(nn.Module):
    def __init__(self, dim, hidden_dim=256, n_layers=4):
        super().__init__()
        self.energy_net = EnergyNetwork(dim, hidden_dim, n_layers)
        self.temperature_schedule = self.get_temperature_schedule()

    def energy(self, x, t):
        """Compute energy U(x,t)"""
        return self.energy_net(x, t)

    def score(self, x, t):
        """Compute thermodynamic score"""
        x.requires_grad_(True)
        energy = self.energy(x, t)
        score = -torch.autograd.grad(
            energy.sum(), x, create_graph=True
        )[0]

        temperature = self.get_temperature(t)
        return score / temperature

    def get_temperature(self, t):
        """Get temperature at time t"""
        return self.temperature_schedule(t)

Reverse Diffusion Process¶

Thermodynamic Reverse SDE¶

The reverse-time SDE with thermodynamic interpretation: \(\(d\mathbf{x} = \left[\mathbf{f}(\mathbf{x}, t) - g(t)^2 s_\theta(\mathbf{x}, t)\right]dt + g(t)d\bar{\mathbf{w}}\)\)

Where:

\(\mathbf{f}(\mathbf{x}, t)\) is drift coefficient
\(g(t)\) is diffusion coefficient
\(s_\theta(\mathbf{x}, t)\) is learned score function
\(d\bar{\mathbf{w}}\) is reverse Wiener process

Heat Equation Connection¶

The reverse process satisfies a modified heat equation: \(\(\frac{\partial p}{\partial t} = \nabla \cdot \left(D(t) \nabla p + D(t) p \nabla \log p_t\right)\)\)

Where \(D(t) = g(t)^2/2\) is diffusion coefficient.

Langevin Dynamics¶

Discrete sampling via Langevin MCMC: \(\(\mathbf{x}_{i+1} = \mathbf{x}_i + \epsilon s_\theta(\mathbf{x}_i, t) + \sqrt{2\epsilon T_t}\boldsymbol{\xi}\)\)

Training Objectives¶

Score Matching¶

Minimize score matching loss: \(\(\mathcal{L}_{\text{SM}} = \mathbb{E}_{t,\mathbf{x}_0,\boldsymbol{\epsilon}}\left[\left\|s_\theta(\mathbf{x}_t, t) - s_t(\mathbf{x}_t)\right\|^2\right]\)\)

Where \(s_t(\mathbf{x}_t)\) is the true score.

Denoising Score Matching¶

Simplified objective using noise prediction: \(\(\mathcal{L}_{\text{DSM}} = \mathbb{E}_{t,\mathbf{x}_0,\boldsymbol{\epsilon}}\left[\left\|\epsilon_\theta(\mathbf{x}_t, t) - \boldsymbol{\epsilon}\right\|^2\right]\)\)

Thermodynamic Consistency Loss¶

Additional term enforcing thermodynamic relations: \(\(\mathcal{L}_{\text{thermo}} = \mathbb{E}\left[\left|U + TS - F\right|^2 + \left|\frac{\partial F}{\partial T} + S\right|^2\right]\)\)

Energy Conservation¶

Penalize energy violations: \(\(\mathcal{L}_{\text{energy}} = \mathbb{E}\left[\left|\frac{dE}{dt} - P_{\text{input}} + P_{\text{dissipation}}\right|^2\right]\)\)

Specialized Architectures¶

Energy-Based Networks¶

Networks that explicitly output energy: \(\(U_\theta(\mathbf{x}, t) = \text{EnergyNet}(\mathbf{x}, t)\)\)

Common architectures:

ResNet-based energy networks
Transformer energy models
Graph neural networks for molecular systems

Temperature-Adaptive Networks¶

Networks with learnable temperature schedules: \(\(T_\theta(t) = \text{TempNet}(t)\)\)

Multi-Scale Models¶

Hierarchical models for different length scales: \(\(U_{\text{total}} = U_{\text{atomic}} + U_{\text{molecular}} + U_{\text{system}}\)\)

Sampling Methods¶

Euler-Maruyama Scheme¶

Basic numerical integration: \(\(\mathbf{x}_{i+1} = \mathbf{x}_i + h \mathbf{f}(\mathbf{x}_i, t_i) + \sqrt{h} g(t_i) \boldsymbol{\xi}_i\)\)

Heun's Method¶

Higher-order accuracy: \(\(\tilde{\mathbf{x}}_{i+1} = \mathbf{x}_i + h \mathbf{f}(\mathbf{x}_i, t_i) + \sqrt{h} g(t_i) \boldsymbol{\xi}_i\)\) \(\(\mathbf{x}_{i+1} = \mathbf{x}_i + \frac{h}{2}[\mathbf{f}(\mathbf{x}_i, t_i) + \mathbf{f}(\tilde{\mathbf{x}}_{i+1}, t_{i+1})] + \sqrt{h} g(t_i) \boldsymbol{\xi}_i\)\)

Predictor-Corrector¶

Combine prediction and correction steps:

Predictor: Standard Euler step
Corrector: Langevin MCMC refinement

Adaptive Sampling¶

Adjust step size based on local dynamics: \(\(h_{i+1} = h_i \cdot \text{adapt\_factor}(\|\mathbf{f}(\mathbf{x}_i, t_i)\|, \text{error\_estimate})\)\)

Temperature Schedules¶

Linear Schedule¶

\[T_t = T_{\text{start}} \frac{T_{\text{end}} - t}{T_{\text{end}} - T_{\text{start}}}\]

Exponential Schedule¶

\[T_t = T_{\text{start}} \exp\left(-\frac{t}{\tau}\right)\]

Cosine Schedule¶

\[T_t = T_{\text{end}} + \frac{T_{\text{start}} - T_{\text{end}}}{2}\left(1 + \cos\left(\frac{\pi t}{T}\right)\right)\]

Learned Schedule¶

\[T_t = \text{ScheduleNet}(t, \text{problem\_features})\]

Physical Constraints¶

Conservation Laws¶

Enforce conservation during generation:

Energy Conservation: \(\(\sum_i E_i = \text{constant}\)\)

Momentum Conservation: \(\(\sum_i m_i \mathbf{v}_i = \text{constant}\)\)

Mass Conservation: \(\(\sum_i m_i = \text{constant}\)\)

Symmetries¶

Respect physical symmetries:

Translation Invariance: \(\(U(\mathbf{x} + \mathbf{a}) = U(\mathbf{x})\)\)

Rotation Invariance: \(\(U(R\mathbf{x}) = U(\mathbf{x})\)\)

Permutation Invariance: \(\(U(P\mathbf{x}) = U(\mathbf{x})\)\)

Boundary Conditions¶

Handle different boundary conditions:

Periodic Boundaries: \(\(\mathbf{x}(L) = \mathbf{x}(0)\)\)

Reflecting Boundaries: \(\(\mathbf{v} \cdot \mathbf{n} = 0\)\) at boundaries

Absorbing Boundaries: \(\(p(\mathbf{x}) = 0\)\) at boundaries

Mixture Models¶

Generate from multiple modes: \(\(p(\mathbf{x}) = \sum_k \pi_k p_k(\mathbf{x})\)\)

Each mode has its own energy function: \(\(U_k(\mathbf{x}) = U_{\text{base}}(\mathbf{x}) + V_k(\mathbf{x})\)\)

Mode Switching¶

Allow transitions between modes during generation: \(\(P(k \to j) = \exp\left(-\frac{U_j - U_k}{k_B T}\right)\)\)

Hierarchical Generation¶

Generate at multiple scales:

Global structure
Local details
Fine-scale features

Conditional Generation¶

Conditional Score Models¶

Score function conditioned on context: \(\(s_\theta(\mathbf{x}, t | \mathbf{c}) = \nabla_{\mathbf{x}} \log p_t(\mathbf{x} | \mathbf{c})\)\)

Classifier Guidance¶

Use external classifier for guidance: \(\(\tilde{s}_\theta(\mathbf{x}, t) = s_\theta(\mathbf{x}, t) + w \nabla_{\mathbf{x}} \log p_{\phi}(y | \mathbf{x})\)\)

Classifier-Free Guidance¶

Self-contained conditional generation: \(\(\tilde{s}_\theta(\mathbf{x}, t) = s_\theta(\mathbf{x}, t | \mathbf{c}) + w(s_\theta(\mathbf{x}, t | \mathbf{c}) - s_\theta(\mathbf{x}, t))\)\)

Applications¶

Molecular Dynamics¶

Generate molecular configurations:

Protein folding trajectories
Chemical reaction pathways
Drug design and optimization

Material Design¶

Generate new materials:

Crystal structures
Polymer configurations
Nanoparticle assemblies

Climate Modeling¶

Generate weather patterns:

Temperature distributions
Precipitation patterns
Extreme event simulations

Fluid Dynamics¶

Generate flow fields:

Turbulent flows
Heat transfer patterns
Multiphase flows

Advanced Techniques¶

Neural ODEs for Diffusion¶

Use neural ODEs for continuous-time modeling: \(\(\frac{d\mathbf{x}}{dt} = f_\theta(\mathbf{x}, t)\)\)

Stochastic Interpolants¶

Learn paths between distributions: \(\(\mathbf{x}_t = (1-t)\mathbf{x}_0 + t\mathbf{x}_1 + \sigma_t \boldsymbol{\epsilon}\)\)

Flow Matching¶

Match vector fields instead of scores: \(\(\mathcal{L}_{\text{FM}} = \mathbb{E}_{t,\mathbf{x}_0,\mathbf{x}_1}\left[\left\|v_\theta(\mathbf{x}_t, t) - u_t(\mathbf{x}_t)\right\|^2\right]\)\)

Evaluation Metrics¶

Thermodynamic Consistency¶

Check thermodynamic relations:

\(dU = TdS - PdV\)
\(G = H - TS\)
Maxwell relations

Sample Quality¶

Standard generative model metrics:

Fréchet Inception Distance (FID)
Inception Score (IS)
Kernel Inception Distance (KID)

Physical Realism¶

Domain-specific validation:

Energy conservation
Force consistency
Stability analysis

Computational Considerations¶

Memory Optimization¶

Techniques for large-scale generation:

Gradient checkpointing
Mixed precision training
Model parallelism

Acceleration Methods¶

Speed up sampling:

Distillation models
Deterministic sampling
Few-step generation

Hardware Optimization¶

Efficient implementation:

GPU optimization
TPU acceleration
Distributed sampling

Future Directions¶

Quantum Diffusion Models¶

Extension to quantum systems: \(\(\frac{\partial \rho}{\partial t} = -\frac{i}{\hbar}[H, \rho] + \mathcal{L}[\rho]\)\)

Non-Equilibrium Diffusion¶

Models for driven systems: \(\(\frac{d\mathbf{x}}{dt} = -\nabla U(\mathbf{x}) + \mathbf{F}_{\text{drive}} + \boldsymbol{\xi}\)\)

Adaptive Neural Architectures¶

Networks that adapt during generation: \(\(\theta_{t+1} = \theta_t + \Delta\theta(\mathbf{x}_t, t)\)\)

Conclusion¶

Thermodynamic diffusion models provide a powerful framework for generating samples that respect physical principles and constraints. By incorporating explicit thermodynamic variables and conservation laws, these models can generate realistic and physically consistent samples across a wide range of applications, from molecular systems to climate modeling.