Stochastic Interpolants

• Stochastic interpolants are mathematical constructs that define a continuous stochastic process accurately linking two probability distributions via deterministic and diffusion dynamics.
• They unify normalizing flows and score-based diffusions using explicit formulations for drift and score functions optimized with quadratic loss objectives.
• They are pivotal in generative modeling, high-dimensional sampling, and forecasting, with extensions to multimarginal, Riemannian, and latent variable settings.

Stochastic interpolants are mathematical constructs and algorithmic frameworks designed to generate a continuous or stochastic process that exactly bridges two prescribed probability distributions in a finite time interval. Originally developed to unify and extend normalizing flows and score-based diffusion models, stochastic interpolants have emerged as a central paradigm in generative modeling, scientific computing, probabilistic forecasting, and high-dimensional sampling. Their flexibility allows the realization of both deterministic transport and stochastic diffusion-based transformations, and modern research has generalized them to multimarginal, Riemannian, data-dependent, and latent-variable settings.

1. General Framework of Stochastic Interpolants

At the core, a stochastic interpolant is a time-indexed family of random variables $\{x_t\}_{t\in[0,1]}$ constructed so that $x_0$ follows an initial (“base”) distribution $\rho_0$, $x_1$ follows a target distribution $\rho_1$, and the path between these endpoints defines an explicit, typically smooth, interpolation in probability space. The canonical form is

$x_t = I(t, x_0, x_1) + \gamma(t) z$

with $x_0 \sim \rho_0$, $x_1 \sim \rho_1$, $z \sim \mathcal{N}(0, I)$, and $I(0, x_0, x_1) = x_0$, $I(1, x_0, x_1) = x_1$, $\gamma(0) = \gamma(1) = 0$ (Albergo et al., 2023). Various choices of $I$ (e.g., linear, trigonometric) and $\gamma(t)$ (deterministic or vanishing at endpoints) allow the specification of the “type” of interpolant: from strictly deterministic (flow-based) to fully stochastic (diffusion-based).

The time-evolving marginal densities $\rho_t$ of $x_t$ satisfy a first-order continuity equation

$$\partial_t \rho(t, x) + \nabla \cdot (b(t, x) \rho(t, x)) = 0,$$

where $b(t, x)$ is the velocity field obtained by conditional expectation of the time derivative given $x_t$: $b(t, x) = \mathbb{E}[\partial_t x_t | x_t = x].$ Including stochasticity introduces a diffusion term so that $\rho_t$ satisfies a (forward or backward) Fokker–Planck equation (Albergo et al., 2023).

These interpolants generalize both deterministic optimal transport (e.g., normalizing flows) and noise-driven generative models (e.g., score-based diffusions) (Albergo et al., 2022). The unifying mathematical structure enables flexible, model-agnostic specification of sampling, transform, and generative operations.

2. Mathematical Formulation and Learning

The functional form of $I(t, x_0, x_1)$, $\gamma(t)$, and the endogenous (or exogenous) noise term $z$ determines the character of the interpolant:

• Linear interpolant: $x_t = (1-t)x_0 + t x_1 + \gamma(t) z$, with $\gamma(t)$ possibly $\sqrt{t(1-t)}$ (Albergo et al., 2023), recovers ordinary flows when $\gamma(t) = 0$ and classical diffusions when $\gamma(t) > 0$.
• Trigonometric or other smooth interpolants: $$x_t = \cos(\frac{\pi}{2} t)x_0 + \sin(\frac{\pi}{2} t)x_1$$ (Albergo et al., 2022).

The drift $b(t, x)$ and the score function $s(t, x) = \nabla \log \rho(t, x)$ are key quantities for constructing transport and sampling algorithms. These are framed as minimizers of explicit quadratic objective functions: $$\mathcal{L}_b[\hat{b}] = \int_0^1 \mathbb{E} \left[ \frac{1}{2}|\hat{b}(t, x_t)|^2 - (\partial_t I(t, x_0, x_1) + \dot{\gamma}(t) z) \cdot \hat{b}(t, x_t) \right] dt$$ and similarly for the score (Albergo et al., 2023, Albergo et al., 2022). These objectives enable training via regression or empirical risk minimization, sidestepping deep simulation-based losses.

Sampling is conducted by integrating the learned drift or joint drift-score fields using forward or reverse ODEs or SDEs:

• Deterministic (ODE/“probability flow”): $dX_t = b(t, X_t) dt$.
• Stochastic (SDE): $$dX_t = b_F(t, X_t) dt + \sqrt{2\varepsilon(t)} dW_t$$, where $b_F(t, x) = b(t, x) + \varepsilon(t) s(t, x)$.

This approach covers both exact sampling (when $b$ and $s$ are learned perfectly) and approximate settings, with theoretical bounds on bias and discretization error (2502.09130, 2504.15736).

3. Generalizations and Extensions

The stochastic interpolant framework has been extended substantially:

• Multimarginal Interpolants: By parameterizing the interpolation with a simplex coordinate $\alpha$ rather than scalar $t$, the method bridges more than two densities simultaneously, yielding a process $x(\alpha) = \sum_{k=0}^K \alpha_k x_k$ over the simplex $\Delta^K$, enabling applications like all-to-all style transfer and algorithmic fairness (Albergo et al., 2023).
• Data-dependent Couplings: By learning or specifying couplings $\rho(x_0, x_1)$ beyond product form, the interpolant can define conditional generative models, improving performance in tasks such as super-resolution and in-painting (Albergo et al., 2023).
• Riemannian Manifolds: For settings like $\mathbb{S}^2$ (sphere), SO(3) (rotations), and general Riemannian manifolds, geodesic-based interpolants $$I(t; x_0, x_1) = \operatorname{Exp}_{x_0}(t \cdot \log_{x_0} x_1)$$ are used, and the marginal flows are governed by transport equations on the manifold (2504.15736).
• Latent Variable Models: Latent Stochastic Interpolants (LSI) define interpolations in learned latent spaces, enabling joint training of encoder, decoder, and SI modules via a continuous-time Evidence Lower Bound (ELBO) (2506.02276).

4. Algorithms and Practical Implementations

Typical implementation of stochastic interpolants involves learning neural parameterizations of $b_\theta(t, x)$ and $s_\theta(t, x)$ via regression on samples $(x_0, x_1, z)$, with losses computed as expectations in $t$ (for instance, equispaced or importance sampled in $[0,1]$) (Albergo et al., 2023, Albergo et al., 2022).

Sampling Strategies:

• Forward ODE/SDE integration: Numerically solve $dX_t = b_\theta(t, X_t) dt$ or its stochastic version.
• Discrete-time approximation: Euler–Maruyama schemes are analyzed with finite-time KL error bounds, with scheduling strategies (e.g., exponentially decaying timestep sizes) designed to control discretization error due to stiff interpolants and small latent noise scales (2502.09130).
• Conditional/Multimarginal models: Learn conditional velocities $g_k(\alpha, x)$ for each marginal and interpolate as $$b(t, x) = \sum_{k=0}^K \dot{\alpha}_k(t) g_k(\alpha(t), x)$$ (Albergo et al., 2023).
• Energy-consistent (Physics-aware) Interpolants: Parameterize $\alpha$, $\beta$ in (possibly Fourier) bases and optimize to preserve physical invariants (such as kinetic energy), crucial for long fluid dynamics rollouts (2504.05852).

Theoretical Analysis:

• Contractivity: For Gaussian to log-concave targets, the SI flow map is Lipschitz with constants matching those from Caffarelli’s theorem for optimal transport. This ensures stability and robustness for sampling and estimation (2504.10653).
• Generative Bias on Manifolds: KL-divergence between generated and target marginal laws depends on mismatches in learned velocity and score fields, with explicit bounds given in terms of time-integrated inner products (2504.15736).

Extension to High Dimensions:

Machine learning approaches—including neural ODEs, denoising networks, or neural FBSDE solvers—allow tractable implementation even for $d \gg 1$ (2502.00355).

5. Applications in Generative Modeling and Scientific Computing

Stochastic interpolants have enabled advances in numerous domains:

• Unifying Flows and Diffusions: The framework is the first to provably and practically connect flow-based and diffusion-based models, allowing exact finite-time mappings between arbitrary base and target densities (Albergo et al., 2023).
• Multimodal and Conditional Generation: Can be applied to multimarginal problems (all-to-all translation, fair generation), and to conditional tasks (super-resolution, in-painting) via data-coupled interpolants (Albergo et al., 2023, Albergo et al., 2023).
• Material and Molecular Generation: Used as the generative core in open-ended material discovery (Open Materials Generation) and Boltzmann sampling (e.g., BoltzNCE), with state-of-the-art accuracy and efficiency (2502.02582, Aggarwal et al., 1 Jul 2025).
• Protein and Fluid Dynamics Simulations: SI-based models (with SO(3)-equivariance or energy-consistency) enable accelerated molecular dynamics and long-horizon, stable fluid simulations, outperforming classical and contemporary deep learning models (2410.09667, 2504.05852).
• Time Series and Forecasting: Stochastic interpolants are combined with recurrence and SDEs for efficient probabilistic forecasting of multivariate time series and high-dimensional physical systems, including FöLLMer (optimal transport) sampling for conditional distributions (2409.11684, Chen et al., 20 Mar 2024).

6. Theoretical and Practical Implications

The interpolant framework offers explicit, simulation-free (quadratic loss) training objectives and has clarified the relationship between transport, diffusion, and score functions. The contractivity and monotonicity properties derived in recent work have enabled precise control of sampling error, stability, and regularity, matching optimal transport and functional inequality bounds (2504.10653, 2502.09130).

Recent extensions to Riemannian manifolds and latent-variable models allow for learning on complex geometric domains and for matching in joint latent-observation spaces, leveraging advanced sampling schemes such as embedding-SDEs and flexible ELBOs (2504.15736, 2506.02276).

Limitations include the potential for numerical instability near endpoints (when the latent noise scale vanishes), increased computational costs in score estimation (for high-dimensional targets), and open questions regarding scalability to very large molecular or materials systems using current neural architectures (Aggarwal et al., 1 Jul 2025). Ongoing research addresses adaptive time stepping, coupling structure, further physical invariants, and broader classes of governing SDEs.

7. Outlook and Future Directions

The stochastic interpolant paradigm is anticipated to underpin the next generation of generative models for scientific, geometric, and structured probabilistic data. Promising directions include enhanced coupling for inverse problems, hybridization with transformer and sequence models for temporal prediction, augmentation with automatic symmetry and constraint incorporation, and deeper integration with control-theoretic methods for stochastic process design (2410.09667, 2502.02582, 2504.15736).

The flexibility and theoretical foundation provided by stochastic interpolants—expressed in mathematical formulations for velocity, score, and transport fields; empirical risk minimization algorithms; and contractivity theorems for sampling—form a robust basis for future unified approaches to high-fidelity generative modeling across statistical, physical, and data-driven settings.