Victor Armegioiu

Roberto Molinaro, Samuel Lanthaler, Bogdan Raonić et al.

We present a generative AI algorithm for addressing the pressing task of fast, accurate, and robust statistical computation of three-dimensional turbulent fluid flows. Our algorithm, termed as GenCFD, is based on an end-to-end conditional score-based diffusion model. Through extensive numerical experimentation with a set of challenging fluid flows, we demonstrate that GenCFD provides an accurate approximation of relevant statistical quantities of interest while also efficiently generating high-quality realistic samples of turbulent fluid flows and ensuring excellent spectral resolution. In contrast, ensembles of deterministic ML algorithms, trained to minimize mean 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 mathematically relevant features of turbulent fluid flows while being amenable to explicit analytical formulae. Our codes are publicly available at https://github.com/camlab-ethz/GenCFD.

Victor Armegioiu, Juan Carrasquilla, Siddhartha Mishra et al.

We propose a framework for the design and analysis of optimization algorithms in variational quantum Monte Carlo, drawing on geometric insights into the corresponding function space. The framework translates infinite-dimensional optimization dynamics into tractable parameter-space algorithms through a Galerkin projection onto the tangent space of the variational ansatz. This perspective unifies existing methods such as stochastic reconfiguration and Rayleigh-Gauss-Newton, provides connections to classic function-space algorithms, and motivates the derivation of novel algorithms with geometrically principled hyperparameter choices. We validate our framework with numerical experiments demonstrating its practical relevance through the accurate estimation of ground-state energies for several prototypical models in condensed matter physics modeled with neural network wavefunctions.

Victor Armegioiu, Yannick Ramic, Siddhartha Mishra

The statistical modeling of fluid flows is very challenging due to their multiscale dynamics and extreme sensitivity to initial conditions. While recently proposed conditional diffusion models achieve high fidelity, they typically require hundreds of stochastic sampling steps at inference. We introduce a rectified flow framework that learns a time-dependent velocity field, transporting input to output distributions along nearly straight trajectories. By casting sampling as solving an ordinary differential equation (ODE) along this straighter flow field, our method makes each integration step much more effective, using as few as eight steps versus (more than) 128 steps in standard score-based diffusion, without sacrificing predictive fidelity. Experiments on challenging multiscale flow benchmarks show that rectified flows recover the same posterior distributions as diffusion models, preserve fine-scale features that MSE-trained baselines miss, and deliver high-resolution samples in a fraction of inference time.

Victor Armegioiu

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.