Remarkably, the solutions learned for each individual task resemble those obtained by solving a kernel regression problem, revealing a novel connection between neural networks and kernel methods.

Stochastic interpolants unify flows and diffusions, popular generative modeling frameworks. A primary hyperparameter in these methods is the interpolation schedule that determines how to bridge a standard Gaussian base measure to an arbitrary target measure. We prove how to convert a sample path of a stochastic differential equation (SDE) with arbitrary diffusion coefficient under any schedule into the unique sample path under another arbitrary schedule and diffusion coefficient. We then extend the stochastic interpolant framework to admit a larger class of point mass schedules in which the Gaussian base measure collapses to a point mass measure. Under the assumption of Gaussian data, we identify lazy schedule families that make the drift identically zero and show that with deterministic sampling one gets a variance-preserving schedule commonly used in diffusion models, whereas with statistically optimal SDE sampling one gets our point mass schedule. Finally, to demonstrate the usefulness of our theoretical results on realistic highly non-Gaussian data, we apply our lazy schedule conversion to a state-of-the-art pretrained flow model and show that this allows for generating images in fewer steps without retraining the model.

We propose a simple, scalable algorithm for using stochastic interpolants to sample from unnormalized densities and for fine-tuning generative models. The approach, Tilt Matching, arises from a dynamical equation relating the flow matching velocity to one targeting the same distribution tilted by a reward, implicitly solving a stochastic optimal control problem. The new velocity inherits the regularity of stochastic interpolant transports while also being the minimizer of an objective with strictly lower variance than flow matching itself. The update to the velocity field can be interpreted as the sum of all joint cumulants of the stochastic interpolant and copies of the reward, and to first order is their covariance. The algorithms do not require any access to gradients of the reward or backpropagating through trajectories of the flow or diffusion. We empirically verify that the approach is efficient and highly scalable, providing state-of-the-art results on sampling under Lennard-Jones potentials and is competitive on fine-tuning Stable Diffusion, without requiring reward multipliers. It can also be straightforwardly applied to tilting few-step flow map models.

Modern generative models hold great promise for accelerating diverse tasks involving the simulation of physical systems, but they must be adapted to the specific constraints of each domain. Significant progress has been made for biomolecules and crystalline materials. Here, we address amorphous materials (glasses), which are disordered particle systems lacking atomic periodicity. Sampling equilibrium configurations of glass-forming materials is a notoriously slow and difficult task. This obstacle could be overcome by developing a generative framework capable of producing equilibrium configurations with well-defined likelihoods. In this work, we address this challenge by leveraging an equivari-ant Riemannian stochastic interpolation framework which combines Riemannian stochastic interpolant and equivariant flow matching. Our method rigorously incorporates periodic boundary conditions and the symmetries of multi-component particle systems, adapting an equivariant graph neural network to operate directly on the torus. Our numerical experiments on model amorphous systems demonstrate that enforcing geometric and symmetry constraints significantly improves generative performance.

Transport-based methods have emerged as a leading paradigm for building generative models from large, clean datasets. However, in many scientific and engineering domains, clean data are often unavailable: instead, we only observe measurements corrupted through a noisy, ill-conditioned channel. A generative model for the original data thus requires solving an inverse problem at the level of distributions. In this work, we introduce a novel approach to this task based on Stochastic Interpolants: we iteratively update a transport map between corrupted and clean data samples using only access to the corrupted dataset as well as black box access to the corruption channel. Under appropriate conditions, this iterative procedure converges towards a self-consistent transport map that effectively inverts the corruption channel, thus enabling a generative model for the clean data. We refer to the resulting method as the self-consistent stochastic interpolant (SCSI). It (i) is computationally efficient compared to variational alternatives, (ii) highly flexible, handling arbitrary nonlinear forward models with only black-box access, and (iii) enjoys theoretical guarantees. We demonstrate superior performance on inverse problems in natural image processing and scientific reconstruction, and establish convergence guarantees of the scheme under appropriate assumptions.

Craig interpolation and uniform interpolation have many applications in knowledge representation, including explainability, forgetting, modularization and reuse, and even learning. At the same time, many relevant knowledge representation formalisms do in general not have Craig or uniform interpolation, and computing interpolants in practice is challenging. We have a closer look at two prominent knowledge representation formalisms, description logics and logic programming, and discuss theoretical results and practical methods for computing interpolants.

Explainability of decision-making AI systems (XAI), and specifically neural networks (NNs), is a key requirement for deploying AI in sensitive areas [18]. A recent trend in explaining NNs is based on formal methods and logic, providing explanations for the decisions of machine learning systems [24, 31, 32, 41, 42, 44] accompanied by provable guarantees regarding their correctness. Yet, rigorous exploration of the continuous feature space requires to estimate decision boundaries with complex shapes. This, however, remains a challenge because existing explanations [24, 31, 32, 41, 42, 44] constrain only individual features and hence fail capturing relationships among the features that are essential to understand the reasons behind the multi-parametrized classification process. We address the need to provide interpretations of NN systems that are as meaningful as possible using a novel concept of Space Explanations, delivered by a flexible symbolic reasoning framework where Craig interpolation [12] is at the heart of the machinery.

Diffusion- and flow-based policies deliver state-of-the-art performance on long-horizon robotic manipulation and imitation learning tasks. However, these controllers employ a fixed inference budget at every control step, regardless of task complexity, leading to computational inefficiency for simple subtasks while potentially underperforming on challenging ones. To address these issues, we introduce Difficulty-Aware Stochastic Interpolant Policy (DA-SIP), a framework that enables robotic controllers to adaptively adjust their integration horizon in real time based on task difficulty. Our approach employs a difficulty classifier that analyzes observations to dynamically select the step budget, the optimal solver variant, and ODE/SDE integration at each control cycle. DA-SIP builds upon the stochastic interpolant formulation to provide a unified framework that unlocks diverse training and inference configurations for diffusion- and flow-based policies. Through comprehensive benchmarks across diverse manipulation tasks, DA-SIP achieves 2.6-4.4x reduction in total computation time while maintaining task success rates comparable to fixed maximum-computation baselines. By implementing adaptive computation within this framework, DA-SIP transforms generative robot controllers into efficient, task-aware systems that intelligently allocate inference resources where they provide the greatest benefit.