Memory-Conditioned Flow-Matching for Stable Autoregressive PDE Rollouts

Autoregressive generative PDE solvers can be accurate one step ahead yet drift over long rollouts, especially in coarse-to-fine regimes where each step must regenerate unresolved fine scales. This is the regime of diffusion and flow-matching generators: although their internal dynamics are Markovian, rollout stability is governed by per-step \emph{conditional law} errors. Using the Mori--Zwanzig projection formalism, we show that eliminating unresolved variables yields an exact resolved evolution with a Markov term, a memory term, and an orthogonal forcing, exposing a structural limitation of memoryless closures. Motivated by this, we introduce memory-conditioned diffusion/flow-matching with a compact online state injected into denoising via latent features. Via disintegration, memory induces a structured conditional tail prior for unresolved scales and reduces the transport needed to populate missing frequencies. We prove Wasserstein stability of the resulting conditional kernel. We then derive discrete Grönwall rollout bounds that separate memory approximation from conditional generation error. Experiments on compressible flows with shocks and multiscale mixing show improved accuracy and markedly more stable long-horizon rollouts, with better fine-scale spectral and statistical fidelity.

Out-of-Distribution Detection in Molecular Complexes via Diffusion Models for Irregular Graphs

Predictive machine learning models generally excel on in-distribution data, but their performance degrades on out-of-distribution (OOD) inputs. Reliable deployment therefore requires robust OOD detection, yet this is particularly challenging for irregular 3D graphs that combine continuous geometry with categorical identities and are unordered by construction. Here, we present a probabilistic OOD detection framework for complex 3D graph data built on a diffusion model that learns a density of the training distribution in a fully unsupervised manner. A key ingredient we introduce is a unified continuous diffusion over both 3D coordinates and discrete features: categorical identities are embedded in a continuous space and trained with cross-entropy, while the corresponding diffusion score is obtained analytically via posterior-mean interpolation from predicted class probabilities. This yields a single self-consistent probability-flow ODE (PF-ODE) that produces per-sample log-likelihoods, providing a principled typicality score for distribution shift. We validate the approach on protein-ligand complexes and construct strict OOD datasets by withholding entire protein families from training. PF-ODE likelihoods identify held-out families as OOD and correlate strongly with prediction errors of an independent binding-affinity model (GEMS), enabling a priori reliability estimates on new complexes. Beyond scalar likelihoods, we show that multi-scale PF-ODE trajectory statistics - including path tortuosity, flow stiffness, and vector-field instability - provide complementary OOD information. Modeling the joint distribution of these trajectory features yields a practical, high-sensitivity detector that improves separation over likelihood-only baselines, offering a label-free OOD quantification workflow for geometric deep learning.

Generative AI for fast and accurate Statistical Computation of Fluids

We present a generative AI algorithm for addressing the challenging task of fast, accurate and robust statistical computation of three-dimensional turbulent fluid flows. Our algorithm, termed as GenCFD, is based on a conditional score-based diffusion model. Through extensive numerical experimentation with both incompressible and compressible fluid flows, we demonstrate that GenCFD provides very accurate approximation of statistical quantities of interest such as mean, variance, point pdfs, higher-order moments, while also generating high quality realistic samples of turbulent fluid flows and ensuring excellent spectral resolution. In contrast, ensembles of operator learning baselines which are trained to minimize mean (absolute) square errors regress to the mean flow. We present rigorous theoretical results uncovering the surprising mechanisms through which diffusion models accurately generate fluid flows. These mechanisms are illustrated with solvable toy models that exhibit the relevant features of turbulent fluid flows while being amenable to explicit analytical formulas.