Jian-Xun Wang

Data assimilation and scientific inverse problems require reconstructing high-dimensional physical states from sparse and noisy observations, ideally with uncertainty-aware posterior samples that remain faithful to learned priors and governing physics. While training-free conditional generation is well developed for diffusion models, corresponding conditioning and posterior sampling strategies for Flow Matching (FM) priors remain comparatively under-explored, especially on scientific benchmarks where fidelity must be assessed beyond measurement misfit. In this work, we study training-free conditional generation for scientific inverse problems under FM priors and organize existing inference-time strategies by where measurement information is injected: (i) guided transport dynamics that perturb sampling trajectories using likelihood information, and (ii) source-distribution inference that performs posterior inference over the source variable while keeping the learned transport fixed. Building on the latter, we propose D-Flow SGLD, a source-space posterior sampling method that augments differentiable source inference with preconditioned stochastic gradient Langevin dynamics, enabling scalable exploration of the source posterior induced by new measurement operators without retraining the prior or modifying the learned FM dynamics. We benchmark representative methods from both families on a hierarchy of problems: 2D toy posteriors, chaotic Kuramoto-Sivashinsky trajectories, and wall-bounded turbulence reconstruction. Across these settings, we quantify trade-offs among measurement assimilation, posterior diversity, and physics/statistics fidelity, and establish D-Flow SGLD as a practical FM-compatible posterior sampler for scientific inverse problems.

Data assimilation and scientific inverse problems require reconstructing high-dimensional physical states from sparse and noisy observations, ideally with uncertainty-aware posterior samples that remain faithful to learned priors and governing physics. While training-free conditional generation is well developed for diffusion models, corresponding conditioning and posterior sampling strategies for Flow Matching (FM) priors remain comparatively under-explored, especially on scientific benchmarks where fidelity must be assessed beyond measurement misfit. In this work, we study training-free conditional generation for scientific inverse problems under FM priors and organize existing inference-time strategies by where measurement information is injected: (i) guided transport dynamics that perturb sampling trajectories using likelihood information, and (ii) source-distribution inference that performs posterior inference over the source variable while keeping the learned transport fixed. Building on the latter, we propose D-Flow SGLD, a source-space posterior sampling method that augments differentiable source inference with preconditioned stochastic gradient Langevin dynamics, enabling scalable exploration of the source posterior induced by new measurement operators without retraining the prior or modifying the learned FM dynamics. We benchmark representative methods from both families on a hierarchy of problems: 2D toy posteriors, chaotic Kuramoto-Sivashinsky trajectories, and wall-bounded turbulence reconstruction. Across these settings, we quantify trade-offs among measurement assimilation, posterior diversity, and physics/statistics fidelity, and establish D-Flow SGLD as a practical FM-compatible posterior sampler for scientific inverse problems.

This article presents an intelligent operator learning approach for the rapid prediction of the thermal field in a block of transistors within an integrated circuit. Phonon Boltzmann transport equation (BTE) simulations were conducted to analyze the thermal field in a single dual-fin field-effect transistor. The nanoscale heat flux profile from the top of the substrate was used as an input to simulate a large substrate containing 100 transistors using the Fourier heat conduction equation. Multiple Fourier simulations were performed with randomly varying on/off switching patterns of the surrounding transistors to assess their impact on the thermal field of the central transistor. The substrate side wall temperature profile of the central transistor, obtained from the Fourier simulations, was then used as a boundary condition for a BTE simulation of the same transistor with fins. The resulting BTE data, consisting of mesh coordinates, sidewall temperature profiles, and temperature at each mesh nodes was used to train and test a deep operator network (DeepONet) capable of predicting the thermal field of any arbitrarily chosen transistor. The trained DeepONet was also tested for a large block of 5000 transistors and effectively predicts the thermal field of any arbitrarily selected transistor. The proposed approach enables the prediction of thermal fields in integrated circuits within seconds and can be readily extended for large-scale circuits containing thousands of transistors. These findings underscore DeepONet’s potential as a highly efficient and scalable tool for rapid thermal field prediction in integrated circuits, facilitating real-time thermal management and optimization. By enabling fast and accurate thermal modeling, this approach bridges the gap between detailed phonon transport physics and large-scale circuit analysis, paving the way for advancements in digital twin technology for semiconductor thermal design.

Reconstructing near-wall turbulence from wall-based measurements is a critical yet inherently ill-posed problem in wall-bounded flows, where limited sensing and spatially heterogeneous flow–wall coupling challenge deterministic estimation strategies. To address this, we introduce a novel generative modelling framework based on conditional flow matching for synthesising instantaneous velocity fluctuation fields from wall observations, with explicit quantification of predictive uncertainty. Our method integrates continuous-time flow matching with a probabilistic forward operator trained using stochastic weight-averaging Gaussian, enabling zero-shot conditional generation without model re-training. We demonstrate that the proposed approach not only recovers physically realistic, statistically consistent turbulence structures across the near-wall region but also effectively adapts to various sensor configurations, including sparse, incomplete and low-resolution wall measurements. The model achieves robust uncertainty-aware reconstruction, preserving flow intermittency and structure even under significantly degraded observability. Compared with classical linear stochastic estimation and deterministic convolutional neural network methods, our stochastic generative learning framework exhibits superior generalisation for unseen realisations under same flow conditions and resilience under measurement sparsity with quantified uncertainty. This work establishes a robust semi-supervised generative modelling paradigm for data-consistent flow reconstruction and lays the foundation for uncertainty-aware, sensor-driven modelling of wall-bounded turbulence.

We investigate conditional neural fields (CNFs), mesh-agnostic, coordinate-based decoders conditioned on a low-dimensional latent, for spatial dimensionality reduction of turbulent flows. CNFs are benchmarked against Proper Orthogonal Decomposition and a convolutional autoencoder within a unified encoding–decoding framework and a common evaluation protocol that explicitly separates in-range (interpolative) from out-of-range (strict extrapolative) testing beyond the training horizon, with identical preprocessing, metrics, and fixed splits across all baselines. We examine three conditioning mechanisms: (i) activation-only modulation (often termed FiLM), (ii) low-rank weight + bias modulation (termed FP), and (iii) last-layer inner-product coupling, and introduce a novel domain-decomposed CNF that localizes complexities. Across representative turbulence datasets (WMLES channel inflow, DNS channel inflow, and wall pressure fluctuations over turbulent boundary layers), CNF-FP achieves the lowest training and in-range testing errors, while CNF-FiLM generalizes best for out-of-range scenarios once moderate latent capacity is available. Domain decomposition significantly improves out-of-range accuracy, especially for the more demanding datasets. The study provides a rigorous, physics-aware basis for selecting conditioning, capacity, and domain decomposition when using CNFs for turbulence compression and reconstruction.