Computational Fluid Dynamics

While Physics-Informed Neural Networks (PINNs) offer a mesh-free approach to solving fluid-flow PDEs, standard point-wise residual minimization suffers from convergence pathologies in topologically complex domains like Triply Periodic Minimal Surfaces (TPMS). The locality bias of point-wise constraints fails to propagate global information through tortuous channels, causing unstable gradients and conservation violations. To address this, we propose the Multi-scale Weak-form PINN (MUSA-PINN), which reformulates Navier-Stokes equation constraints as integral conservation laws over hierarchical spherical control volumes. We enforce continuity and momentum conservation via flux-balance residuals on control surfaces. Our method utilizes a three-scale subdomain strategy-comprising large volumes for long-range coupling, skeleton-aware meso-scale volumes aligned with transport pathways, and small volumes for local refinement-alongside a two-stage training schedule prioritizing continuity. Experiments on steady incompressible flow in TPMS geometries show MUSA-PINN outperforms state-of-the-art baselines, reducing relative errors by up to 93% and preserving mass conservation.

Accurate and predictive scale-resolving simulations of laser-ignited rocket engines are highly time-consuming because the problem includes turbulent fuel-oxidizer mixing dynamics, laser-induced energy deposition, and high-speed flame growth. This is conflated with the large design space primarily corresponding to the laser operating conditions and target location. To enable rapid exploration and uncertainty quantification, we propose a data-driven surrogate modeling approach that combines convolutional autoencoders (cAEs) with neural ordinary differential equations (neural ODEs). The present target application of an ML-based surrogate model to leading-edge multi-physics turbulence simulation is part of a paradigm shift in the deployment of surrogate models towards increasing real-world complexity. Sequentially, the cAE spatially compresses high-dimensional flow fields into a low-dimensional latent space, wherein the system's temporal dynamics are learned via neural ODEs. Once trained, the model generates fast spatiotemporal predictions from initial conditions and specified operating inputs. By learning a surrogate to replace the entirety of the time-evolving simulation, the cost of predicting an ignition trial is reduced by several orders of magnitude, allowing efficient exploration of the input parameter space. Further, as the current framework yields a spatiotemporal field prediction, appraisal of the model output's physical grounding is more tractable. This approach marks a significant step toward real-time digital twins for laser-ignited rocket combustors and represents surrogate modeling in a complex system context.

Solving flow through porous media is a crucial step in the topology optimisation of cold plates, a key component in modern thermal management. Traditional computational fluid dynamics (CFD) methods, while accurate, are often prohibitively expensive for large and complex geometries. In contrast, data-driven surrogate models provide a computationally efficient alternative, enabling rapid and reliable predictions. In this study, we develop a machine-learning framework for predicting steady-state flow through porous media governed by the Navier-Stokes-Brinkman equations. We implement and compare three model architectures-convolutional autoencoder (AE), U-Net, and Fourier Neural Operator (FNO)-evaluating their predictive performance. To enhance physics consistency, we incorporate physics-informed loss functions. Our results demonstrate that FNO outperforms AE and U-Net, achieving a mean squared error (MSE) as low as 0.0017 while providing speedups of up to 1000 times compared to CFD. Additionally, the mesh-invariant property of FNO emphasizes its suitability for topology optimisation tasks, where varying mesh resolutions are required. This study highlights the potential of machine learning to accelerate fluid flow predictions in porous media, offering a scalable alternative to traditional numerical methods.

Data-driven surrogate models can significantly accelerate the simulation of continuous dynamical systems, yet the step-wise accumulation of errors during autoregressive time-stepping often leads to spectral blow-up and unphysical divergence. Existing global regularization techniques can enforce contractive dynamics but uniformly damp high-frequency features, causing over-smoothing; meanwhile, long-horizon trajectory optimization methods are severely constrained by memory bottlenecks. This paper proposes Jacobian-Adaptive Weighting for Stability (JAWS), which reformulates operator learning as a Maximum A Posteriori (MAP) estimation problem with spatially heteroscedastic uncertainty, enabling the regularization strength to adapt automatically based on local physical complexity: enforcing contraction in smooth regions to suppress noise while relaxing constraints near singular features such as shocks to preserve gradient information. Experiments demonstrate that JAWS serves as an effective spectral pre-conditioner for trajectory optimization, allowing short-horizon, memory-efficient training to match the accuracy of long-horizon baselines. Validations on the 1D viscous Burgers' equation and 2D flow past a cylinder ($\text{Re}=400$ out-of-distribution generalization) confirm the method's advantages in long-term stability, preservation of physical conservation properties, and computational efficiency. This significant reduction in memory usage makes the method particularly well-suited for stable and efficient long-term simulation of large-scale flow fields in practical engineering applications.

Vision transformers have shown outstanding performance in image generation, yet their adoption in fluid dynamics remains limited. We introduce the Latent Attention on Masked Patches (LAMP) model, an interpretable regression-based modified vision transformer designed for masked flow reconstruction. LAMP follows a three-fold strategy: (i) partition of each flow snapshot into patches, (ii) patch-wise dimensionality reduction via proper orthogonal decomposition, and (iii) reconstruction of the full field from a masked input using a single-layer transformer trained via closed-form linear regression. We test the method on two canonical 2D unsteady wakes: a laminar wake past a bluff body, and a chaotic wake past two cylinders. On the laminar case, LAMP accurately reconstructs the full flow field from a 90%-masked and noisy input, across signal-to-noise ratios between 10 and 30dB. Further, the learned attention matrix yields interpretable multi-fidelity optimal sensor-placement maps. LAMP's performance on the chaotic wake is limited, but outperforms other regression methods such as gappy POD. The modularity of the framework, however, naturally accommodates nonlinear compression and deep attention blocks, thereby providing an efficient baseline for nonlinear, high-dimensional masked flow reconstruction.

The incompressible magnetohydrodynamic (MHD) equations are fundamental in many scientific and engineering applications. However, their strong nonlinearity and dual divergence-free constraints make them highly challenging for conventional numerical solvers. To overcome these difficulties, we propose a Structure-Preserving Randomized Neural Network (SP-RaNN) that automatically and exactly satisfies the divergence-free conditions. Unlike deep neural network (DNN) approaches that rely on expensive nonlinear and nonconvex optimization, SP-RaNN reformulates the training process into a linear least-squares system, thereby eliminating nonconvex optimization. The method linearizes the governing equations through Picard or Newton iterations, discretizes them at collocation points within the domain and on the boundaries using finite-difference schemes, and solves the resulting linear system via a linear least-squares procedure. By design, SP-RaNN preserves the intrinsic mathematical structure of the equations within a unified space-time framework, ensuring both stability and accuracy. Numerical experiments on the Navier-Stokes, Maxwell, and MHD equations demonstrate that SP-RaNN achieves higher accuracy, faster convergence, and exact enforcement of divergence-free constraints compared with both traditional numerical methods and DNN-based approaches. This structure-preserving framework provides an efficient and reliable tool for solving complex PDE systems while rigorously maintaining their underlying physical laws.

State-of-the-art deep learning models have been extensively utilized to reconstruct small-scale structures from coarse-grained data in turbulent flows. However, their application has predominantly been restricted to structured uniform meshes, limiting their applicability to data associated with complex geometries that are typically simulated on structured non-uniform or unstructured meshes. Machine learning (ML) models based on graph neural networks (GNNs), known for their ability to process unstructured data, offer a promising alternative. In this study, we leverage the inherent flexibility of GNNs featuring message passing layers to develop a methodology for reconstructing unresolved small-scale structures from low-resolution data on complex meshes. The accuracy of the proposed approach is demonstrated using two cases: a reacting channel flow on a structured non-uniform mesh, and a reacting hydrogen fueled internal combustion (IC) engine featuring an unstructured mesh. Evaluation of results based on visual agreement, statistical metrics, and cumulative error reduction indicates the effectiveness of the method in accurately reconstructing fine-scale features. Overall, this study provides a pathway for integrating data-driven small-scale reconstruction and subgrid-scale modeling to enhance the accuracy of coarse-grained simulations on complex meshes.

A data-free finite-volume convolutional neural network (FV-CNN) is formulated for steady Euler flow around a cylinder at eight Mach numbers, 2--15. Shared-face fluxes define conservative spatial balances, while automatic differentiation supplies parameter gradients. The derivative mismatch in the earlier automatic-differentiation-based convolutional neural network (AD-CNN) is derived. No computational fluid dynamics (CFD) or experimental field targets enter training; reference-independent selection of coefficients, seeds, and checkpoints is not established. Billig-based upstream fixing conditions shock-location agreement.For eight adopted fields, density and pressure relative $L_2$ discrepancies from finite-resolution CFD are $5.43$--$18.69\%$ and $6.25$--$19.70\%$. Closed-band defects are smaller than AD-CNN in 32/32 self-centered and 22--31/32 CFD-centered comparisons over seven half-widths, 0.06--0.12 cylinder diameters. AD-CNN retains smaller enthalpy root-mean-square (RMS) discrepancies in 7/8 cases, shock-position RMS at Mach 11 and 15, and temperature and transverse-velocity discrepancies at Mach 3.Ten new Mach-15 initializations and adopted seed 42 yield density discrepancies of $12.04$--$43.70\%$ (mean $24.34\%$, sample standard deviation $11.78$ percentage points); the new ten alone give $25.04\%$ and $12.17$ points. Dispersion exceeds the historical adopted-field gap of $3.40$ points; six of eleven outcomes are below AD-CNN. Transition-width ratios of AD-CNN to FV-CNN remain $4.07$--$4.66$, and all 44 self-centered seed--component balances at half-width 0.10 cylinder diameters are smaller for FV-CNN. Local sharpness and this balance metric persist, while initialization-robust accuracy superiority is not established.Component interventions remain one-seed conditional responses; Mach 2 uses a separate budget-limited protocol.

Fluid-solid interaction (FSI) problems are fundamental in many scientific and engineering applications, yet effectively capturing the highly nonlinear two-way interactions remains a significant challenge. Most existing deep learning methods are limited to simplified one-way FSI scenarios, often assuming rigid and static solid to reduce complexity. Even in two-way setups, prevailing approaches struggle to capture dynamic, heterogeneous interactions due to the lack of cross-domain awareness. In this paper, we introduce Fisale, a data-driven framework for handling complex two-way FSI problems. It is inspired by classical numerical methods, namely the Arbitrary Lagrangian-Eulerian (ALE) method and the partitioned coupling algorithm. Fisale explicitly models the coupling interface as a distinct component and leverages multiscale latent ALE grids to provide unified, geometry-aware embeddings across domains. A partitioned coupling module (PCM) further decomposes the problem into structured substeps, enabling progressive modeling of nonlinear interdependencies. Compared to existing models, Fisale introduces a more flexible framework that iteratively handles complex dynamics of solid, fluid and their coupling interface on a unified representation, and enables scalable learning of complex two-way FSI behaviors. Experimentally, Fisale excels in three reality-related challenging FSI scenarios, covering 2D, 3D and various tasks. The code is available at \href{https://github.com/therontau0054/Fisale}.

Reconstructing fluid flows from sparse sensor measurements is a fundamental challenge in science and engineering. Widely separated measurements and complex, multiscale dynamics make accurate recovery of fine-scale structures difficult. In addition, existing methods face a persistent tradeoff: high-accuracy models are often computationally expensive, whereas faster approaches typically compromise fidelity. In this work, we introduce BLISSNet, a model that strikes a strong balance between reconstruction accuracy and computational efficiency for both flow reconstruction and nudging-based data assimilation. The model follows a DeepONet-like architecture, enabling zero-shot inference on domains of arbitrary size. After the first model call on a given domain, certain network components can be precomputed, leading to low inference cost for subsequent evaluations on large domains. Consequently, the model can achieve faster inference than classical interpolation methods such as radial basis function or bicubic interpolation. This combination of high accuracy, low cost, and zero-shot generalization makes BLISSNet well-suited for large-scale real-time flow reconstruction and data assimilation tasks.

Flow physics and more broadly physical phenomena governed by partial differential equations (PDEs), are inherently continuous, high-dimensional and often chaotic in nature. Traditionally, researchers have explored these rich spatiotemporal PDE solution spaces using laboratory experiments and/or computationally expensive numerical simulations. This severely limits automated and large-scale exploration, unlike domains such as drug discovery or materials science, where discrete, tokenizable representations naturally interface with large language models. We address this by coupling multi-agent LLMs with latent foundation models (LFMs), a generative model over parametrised simulations, that learns explicit, compact and disentangled latent representations of flow fields, enabling continuous exploration across governing PDE parameters and boundary conditions. The LFM serves as an on-demand surrogate simulator, allowing agents to query arbitrary parameter configurations at negligible cost. A hierarchical agent architecture orchestrates exploration through a closed loop of hypothesis, experimentation, analysis and verification, with a tool-modular interface requiring no user support. Applied to flow past tandem cylinders at Re = 500, the framework autonomously evaluates over 1,600 parameter-location pairs and discovers divergent scaling laws: a regime-dependent two-mode structure for minimum displacement thickness and a robust linear scaling for maximum momentum thickness, with both landscapes exhibiting a dual-extrema structure that emerges at the near-wake to co-shedding regime transition. The coupling of the learned physical representations with agentic reasoning establishes a general paradigm for automated scientific discovery in PDE-governed systems.

Data assimilation (DA) for compressible flows with shocks is challenging because many classical DA methods generate spurious oscillations and nonphysical features near uncertain shocks. We focus here on the ensemble Kalman filter (EnKF). We show that the poor performance of the EnKF may be attributed to the bimodal forecast distribution that can arise in the vicinity of an uncertain shock location; this violates the assumptions underpinning the EnKF, which assume a forecast which is close to Gaussian. To address this issue we introduce the new neural EnKF. The basic idea is to systematically embed neural function approximations within ensemble DA by mapping the forecast ensemble of shocked flows to the parameter space (weights and biases) of a deep neural network (NN) and to subsequently perform DA in that space. The nonlinear mapping encodes sharp and smooth flow features in an ensemble of NN parameters. Neural EnKF updates are therefore well-behaved only if the NN parameters vary smoothly within the neural representation of the forecast ensemble. We show that such a smooth variation of network parameters can be enforced via physics-informed transfer learning, and demonstrate that in so-doing the neural EnKF avoids the spurious oscillations and nonphysical features that plague the EnKF. The applicability of the neural EnKF is demonstrated through a series of systematic numerical experiments with the inviscid Burgers' equation, the Sod shock tube, and a two-dimensional blast wave.

Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work introduces a physics-informed training framework that addresses these limitations by decomposing PDEs using operator splitting methods, training separate neural operators to learn individual non-linear physical operators while approximating linear operators with fixed finite-difference convolutions. This modular mixture-of-experts architecture enables generalisation to novel physical regimes by explicitly encoding the underlying operator structure. We formulate the modelling task as a neural ordinary differential equation (ODE) where these learned operators constitute the right-hand side, enabling continuous-in-time predictions through standard ODE solvers and implicitly enforcing PDE constraints. Demonstrated on incompressible and compressible Navier--Stokes equations, our approach achieves better convergence and superior performance when generalising to unseen physics. The method remains parameter-efficient, enabling temporal extrapolation beyond training horizons, and provides interpretable components whose behaviour can be verified against known physics.

Modelling rock-fluid interaction requires solving a set of partial differential equations (PDEs) to predict the flow behaviour and the reactions of the fluid with the rock on the interfaces. Conventional high-fidelity numerical models require a high resolution to obtain reliable results, resulting in huge computational expense. This restricts the applicability of these models for multi-query problems, such as uncertainty quantification and optimisation, which require running numerous scenarios. As a cheaper alternative to high-fidelity models, this work develops eight surrogate models for predicting the fluid flow in porous media. Four of these are reduced-order models (ROM) based on one neural network for compression and another for prediction. The other four are single neural networks with the property of grid-size invariance; a term which we use to refer to image-to-image models that are capable of inferring on computational domains that are larger than those used during training. In addition to the novel grid-size-invariant framework for surrogate models, we compare the predictive performance of UNet and UNet++ architectures, and demonstrate that UNet++ outperforms UNet for surrogate models. Furthermore, we show that the grid-size-invariant approach is a reliable way to reduce memory consumption during training, resulting in good correlation between predicted and ground-truth values and outperforming the ROMs analysed. The application analysed is particularly challenging because fluid-induced rock dissolution results in a non-static solid field and, consequently, it cannot be used to help in adjustments of the future prediction.

Learning PDE dynamics for fluids increasingly relies on neural operators and Transformer-based models, yet these approaches often lack interpretability and struggle with localized, high-frequency structures while incurring quadratic cost in spatial samples. We propose representing fields with a Gaussian basis, where learned atoms carry explicit geometry (centers, anisotropic scales, weights) and form a compact, mesh-agnostic, directly visualizable state. Building on this representation, we introduce a Gaussian Particle Operator that acts in modal space: learned Gaussian modal windows perform a Petrov-Galerkin measurement, and PG Gaussian Attention enables global cross-scale coupling. This basis-to-basis design is resolution-agnostic and achieves near-linear complexity in N for a fixed modal budget, supporting irregular geometries and seamless 2D-to-3D extension. On standard PDE benchmarks and real datasets, our method attains state-of-the-art competitive accuracy while providing intrinsic interpretability.

Existing image SR and generic diffusion models transfer poorly to fluid SR: they are sampling-intensive, ignore physical constraints, and often yield spectral mismatch and spurious divergence. We address fluid super-resolution (SR) with ReMD (Residual-Multigrid Diffusion), a physics-consistent diffusion framework. At each reverse step, ReMD performs a multigrid residual correction: the update direction is obtained by coupling data consistency with lightweight physics cues and then correcting the residual across scales; the multiscale hierarchy is instantiated with a multi-wavelet basis to capture both large structures and fine vortical details. This coarse-to-fine design accelerates convergence and preserves fine structures while remaining equation-free. Across atmospheric and oceanic benchmarks, ReMD improves accuracy and spectral fidelity, reduces divergence, and reaches comparable quality with markedly fewer sampling steps than diffusion baselines. Our results show that enforcing physics consistency inside the diffusion process via multigrid residual correction and multi-wavelet multiscale modeling is an effective route to efficient fluid SR. Our code are available on https://github.com/lizhihao2022/ReMD.

Reliable prediction of hydraulic performance is challenging for Piano Key Weir (PKW) design because discharge capacity depends on three-dimensional geometry and operating conditions. Surrogate models can accelerate hydraulic-structure design, but progress is limited by scarce large, well-documented datasets that jointly capture geometric variation, operating conditions, and functional performance. This study presents WeirNet, a large 3D CFD benchmark dataset for geometric surrogate modeling of PKWs. WeirNet contains 3,794 parametric, feasibility-constrained rectangular and trapezoidal PKW geometries, each scheduled at 19 discharge conditions using a consistent free-surface OpenFOAM workflow, resulting in 71,387 completed simulations that form the benchmark and with complete discharge coefficient labels. The dataset is released as multiple modalities compact parametric descriptors, watertight surface meshes and high-resolution point clouds together with standardized tasks and in-distribution and out-of-distribution splits. Representative surrogate families are benchmarked for discharge coefficient prediction. Tree-based regressors on parametric descriptors achieve the best overall accuracy, while point- and mesh-based models remain competitive and offer parameterization-agnostic inference. All surrogates evaluate in milliseconds per sample, providing orders-of-magnitude speedups over CFD runtimes. Out-of-distribution results identify geometry shift as the dominant failure mode compared to unseen discharge values, and data-efficiency experiments show diminishing returns beyond roughly 60% of the training data. By publicly releasing the dataset together with simulation setups and evaluation pipelines, WeirNet establishes a reproducible framework for data-driven hydraulic modeling and enables faster exploration of PKW designs during the early stages of hydraulic planning.

Accurately modeling the spatio-temporal dynamics of blast wave propagation remains a longstanding challenge due to its highly nonlinear behavior, sharp gradients, and burdensome computational cost. While machine learning-based surrogate models offer fast inference as a promising alternative, they suffer from degraded accuracy, particularly evaluated on complex urban layouts or out-of-distribution scenarios. Moreover, autoregressive prediction strategies in such models are prone to error accumulation over long forecasting horizons, limiting their robustness for extended-time simulations. To address these limitations, we propose RGD-Blast, a robust and generalizable deep surrogate model for high-fidelity, long-term blast wave forecasting. RGD-Blast incorporates a multi-scale module to capture both global flow patterns and local boundary interactions, effectively mitigating error accumulation during autoregressive prediction. We introduce a dynamic-static feature coupling mechanism that fuses time-varying pressure fields with static source and layout features, thereby enhancing out-of-distribution generalization. Experiments demonstrate that RGD-Blast achieves a two-order-of-magnitude speedup over traditional numerical methods while maintaining comparable accuracy. In generalization tests on unseen building layouts, the model achieves an average RMSE below 0.01 and an R2 exceeding 0.89 over 280 consecutive time steps. Additional evaluations under varying blast source locations and explosive charge weights further validate its generalization, substantially advancing the state of the art in long-term blast wave modeling.

Physics-governed models are increasingly paired with machine learning for accelerated predictions, yet most "physics--informed" formulations treat the governing equations as a penalty loss whose scale and meaning are set by heuristic balancing. This blurs operator structure, thereby confounding solution approximation error with governing-equation enforcement error and making the solving and learning progress hard to interpret and control. Here we introduce the Neural Basis Method, a projection-based formulation that couples a predefined, physics-conforming neural basis space with an operator-induced residual metric to obtain a well-conditioned deterministic minimization. Stability and reliability then hinge on this metric: the residual is not merely an optimization objective but a computable certificate tied to approximation and enforcement, remaining stable under basis enrichment and yielding reduced coordinates that are learnable across parametric instances. We use advective multiscale Darcian dynamics as a concrete demonstration of this broader point. Our method produce accurate and robust solutions in single solves and enable fast and effective parametric inference with operator learning.

We present a generative modeling framework for synthesizing physically feasible two-dimensional incompressible flows under arbitrary obstacle geometries and boundary conditions. Whereas existing diffusion-based flow generators either ignore physical constraints, impose soft penalties that do not guarantee feasibility, or specialize to fixed geometries, our approach integrates three complementary components: (1) a boundary-conditioned diffusion model operating on velocity fields; (2) a physics-informed training objective incorporating a divergence penalty; and (3) a projection-constrained reverse diffusion process that enforces exact incompressibility through a geometry-aware Helmholtz-Hodge operator. We derive the method as a discrete approximation to constrained Langevin sampling on the manifold of divergence-free vector fields, providing a connection between modern diffusion models and geometric constraint enforcement in incompressible flow spaces. Experiments on analytic Navier-Stokes data and obstacle-bounded flow configurations demonstrate significantly improved divergence, spectral accuracy, vorticity statistics, and boundary consistency relative to unconstrained, projection-only, and penalty-only baselines. Our formulation unifies soft and hard physical structure within diffusion models and provides a foundation for generative modeling of incompressible fields in robotics, graphics, and scientific computing.

Diffusion models recently developed for generative AI tasks can produce high-quality samples while still maintaining diversity among samples to promote mode coverage, providing a promising path for learning stochastic closure models. Compared to other types of generative AI models, such as GANs and VAEs, the sampling speed is known as a key disadvantage of diffusion models. By systematically comparing transport-based generative models on a numerical example of 2D Kolmogorov flows, we show that flow matching in a lower-dimensional latent space is suited for fast sampling of stochastic closure models, enabling single-step sampling that is up to two orders of magnitude faster than iterative diffusion-based approaches. To control the latent space distortion and thus ensure the physical fidelity of the sampled closure term, we compare the implicit regularization offered by a joint training scheme against two explicit regularizers: metric-preserving (MP) and geometry-aware (GA) constraints. Besides offering a faster sampling speed, both explicitly and implicitly regularized latent spaces inherit the key topological information from the lower-dimensional manifold of the original complex dynamical system, which enables the learning of stochastic closure models without demanding a huge amount of training data.

Purpose of Review Imaging derived fractional flow reserve (FFR) is rapidly evolving beyond conventional computational fluid dynamics (CFD) based pipelines toward machine learning (ML), deep learning (DL), and physics informed approaches that enable fast, wire free, and scalable functional assessment of coronary artery stenosis. This review synthesizes recent advances in computed tomography (CT)- and angiography-based FFR measurement, with particular emphasis on emerging physics-informed neural networks and neural operators (PINNs and PINOs), as well as key considerations for their clinical translation. Recent Findings ML/DL approaches have markedly improved automation and computational speed, enabling prediction of pressure and FFR from anatomical descriptors or angiographic contrast dynamics. However, their real-world performance and generalizability can remain variable and sensitive to domain shift, due to multi-center heterogeneity, interpretability challenges, and differences in acquisition protocols and image quality. Physics informed learning introduces conservation structure and boundary condition consistency into model training, improving generalizability and reducing dependence on dense supervision while maintaining rapid inference. Recent evaluation trends increasingly highlight deployment oriented metrics, including calibration, uncertainty quantification, and quality control gatekeeping, as essential for safe clinical use. Summary The field is converging toward imaging derived FFR methods that are faster, more automated, and more reliable. While ML/DL offers substantial efficiency gains, physics informed frameworks such as PINNs and PINOs may provide a more robust balance between speed and physical consistency. Prospective multi center validation and standardized evaluation will be critical to support broad and safe clinical adoption.

Spatiotemporal flows govern diverse phenomena across physics, biology, and engineering, yet modelling their multiscale dynamics remains a central challenge. Despite major advances in physics-informed machine learning, existing approaches struggle to simultaneously maintain long-term temporal evolution and resolve fine-scale structure across chaotic, turbulent, and physiological regimes. Here, we introduce Uni-Flow, a unified autoregressive-diffusion framework that explicitly separates temporal evolution from spatial refinement for modelling complex dynamical systems. The autoregressive component learns low-resolution latent dynamics that preserve large-scale structure and ensure stable long-horizon rollouts, while the diffusion component reconstructs high-resolution physical fields, recovering fine-scale features in a small number of denoising steps. We validate Uni-Flow across canonical benchmarks, including two-dimensional Kolmogorov flow, three-dimensional turbulent channel inflow generation with a quantum-informed autoregressive prior, and patient-specific simulations of aortic coarctation derived from high-fidelity lattice Boltzmann hemodynamic solvers. In the cardiovascular setting, Uni-Flow enables task-level faster than real-time inference of pulsatile hemodynamics, reconstructing high-resolution pressure fields over physiologically relevant time horizons in seconds rather than hours. By transforming high-fidelity hemodynamic simulation from an offline, HPC-bound process into a deployable surrogate, Uni-Flow establishes a pathway to faster-than-real-time modelling of complex multiscale flows, with broad implications for scientific machine learning in flow physics.

We present a novel property-preserving kernel-based operator learning method for incompressible flows governed by the incompressible Navier--Stokes equations. Traditional numerical solvers incur significant computational costs to respect incompressibility. Operator learning offers efficient surrogate models, but current neural operators fail to exactly enforce physical properties such as incompressibility, periodicity, and turbulence. Our kernel method maps input functions to expansion coefficients of output functions in a property-preserving kernel basis, ensuring that predicted velocity fields analytically and simultaneously preserve the aforementioned physical properties. We present universal approximation results and worst-case a priori convergence rates for our framework; empirically, the observed convergence rates exceed the pessimistic predictions across most benchmarks, motivating a formulation of more optimistic convergence rates. Another central contribution of this work is a novel computational framework revolving around streaming construction of kernel Gramians and recursive Schur-complement algorithms to solve the large block linear systems arising from property-preserving kernel methods. This kernel-based framework makes operator learning practical at scales up to $10{,}000$ training functions sampled at up to $10{,}000$ spatial locations. We evaluate the method on challenging 2D and 3D, laminar and turbulent, incompressible flow problems. Our method achieves up to six orders of magnitude lower relative $\ell_2$ errors upon generalization and trains up to five orders of magnitude faster compared to neural operators. Moreover, while our method enforces incompressibility analytically, neural operators exhibit large divergence errors. Our results show that our method provides an accurate and efficient surrogate for incompressible flows.

Adjoint-based shape optimization of ship hulls is a powerful tool for addressing high-dimensional design problems in naval architecture, particularly in minimizing the ship resistance. However, its application to vessels that employ complex propulsion systems introduces significant challenges. They arise from the need for transient simulations extending over long periods of time with small time steps and from the reverse temporal propagation of the primal and adjoint solutions. These challenges place considerable demands on the required storage and computing power, which significantly hamper the use of adjoint methods in the industry. To address this issue, we propose a machine learning-assisted optimization framework that employs a Conditional Variational Autoencoder-based surrogate model of the propulsion system. The surrogate model replicates the time-averaged flow field induced by a Voith Schneider Propeller and replaces the geometrically and time-resolved propeller with a data-driven approximation. Primal flow verification examples demonstrate that the surrogate model achieves significant computational savings while maintaining the necessary accuracy of the resolved propeller. Optimization studies demonstrate that neglecting the propulsion system can result in hull designs whose performance is inferior to that of the initial shape when subsequently validated using a numerically resolved propulsor. In contrast, the proposed method produces shapes that actually achieve more than an 8% reduction in resistance.

Generating dense physical fields from sparse measurements is a fundamental question in sampling, signal processing, and many other applications. State-of-the-art approaches to this problem either rely on spatial statistics that ignore the governing physics, integrate the physics into a multiple-objective optimization process, or require examples of the complete, fully-resolved simulation state during training, which are frequently unavailable outside of synthetic benchmarks. Here, we present a novel alternative that leverages recent advances in the integration of numerical simulators with data-driven models. Namely, we propose a hybrid modeling pipeline that couples Radial Basis Function (RBF) reconstruction with a Neural Network (NN) correction and a Partial Differential Equation (PDE) solver, so that the numerical simulator itself is embedded directly in the training loop of the learned component. Notably, the NN is trained without assuming availability of examples of the fully-resolved simulation state. This is made possible by implementing the PDE solver so that it is end-to-end differentiable, allowing gradients to be backpropagated through the simulation step during training. This grey-box methodology is evaluated on three standard benchmarks from fluid mechanics, where it achieves superior results over statistical and machine-learning-based reconstruction methods.

Urban wind flow reconstruction is essential for assessing air quality, heat dispersion, and pedestrian comfort, yet remains challenging when only sparse sensor data are available. We propose GenDA, a generative data assimilation framework that reconstructs high-resolution wind fields on unstructured meshes from limited observations. The model employs a multiscale graph-based diffusion architecture trained on computational fluid dynamics (CFD) simulations and interprets classifier-free guidance as a learned posterior reconstruction mechanism: the unconditional branch learns a geometry-aware flow prior, while the sensor-conditioned branch injects observational constraints during sampling. This formulation enables obstacle-aware reconstruction and generalization to held-out mesh geometries, wind directions, and sensor configurations within the studied urban-flow setting, without retraining. We consider both sparse fixed sensors and trajectory-based observations using the same reconstruction procedure. When evaluated against supervised graph neural network (GNN) baselines and classical reduced-order data assimilation methods, GenDA reduces the relative root-mean-square error (RRMSE) by 25-57% and increases the structural similarity index (SSIM) by 23-33% across the tested meshes. Experiments are conducted on Reynolds-averaged Navier-Stokes (RANS) simulations of a real urban neighborhood in Bristol, United Kingdom, at a characteristic Reynolds number of $\mathrm{Re}\approx2\times10^{7}$, featuring complex building geometry and irregular terrain. The proposed framework provides a scalable path toward generative, geometry-aware data assimilation for environmental monitoring in complex domains.

We present GeoTransolver, a multiscale geometry-aware physics attention transformer for Computer Aided Engineering (CAE). GeoTransolver extends the Transolver backbone with GALE (Geometry-Aware Latent Embeddings) attention, which pairs physics-aware self-attention on learned state slices with cross-attention to a shared geometry and global context computed via multi-scale ball queries (inspired by Domino) and reused in every block. Implemented and released in NVIDIA PhysicsNeMo, GeoTransolver persistently projects geometry and global parameters, into physical state spaces to anchor computations to domain structure and operating regimes. We benchmark on DrivAerML, SHIFT-SUV, and SHIFT-Wing against Domino, Transolver (PhysicsNeMo implementation), and literature-reported AB-UPT, evaluating drag/lift R2 and relative L1 errors on field variables. As an additional nonlinear structural mechanics application, we also report Transolver and GeoTransolver results on bumper-beam and full-vehicle Body-in-White (BIW) crash-dynamics benchmarks, evaluating relative L2 trajectory error and probe-level kinematic MSE. GeoTransolver delivers improved accuracy, robustness to geometry and regime shifts, and favorable data efficiency; we include DrivAerML ablations and qualitative contour and design-trend results, advancing operator learning for high-fidelity surrogates on complex, irregular, non-linear domains.

Modeling and controlling fluids is critical across science and engineering. Effective flow control can increase lift, reduce drag, enhance mixing, and attenuate noise, potentially unlocking new technologies. Yet controlling fluids is hard: the dynamics are high-dimensional, nonlinear, and multiscale. While reinforcement learning (RL) has recently succeeded in robotics and protein folding through shared benchmarks, fluid dynamics has resisted such progress: each controller is typically tuned to a single geometry and operating point, making results hard to accumulate, transfer, and compare. We introduce HydroGym, a solver-independent RL platform for flow control, and show that standardized infrastructure unlocks transferable control intelligence across flow regimes. HydroGym provides 61+ validated environments spanning laminar to turbulent flows, with systematic Reynolds number progressions up to Re=400,000 and Mach number variations in 2D and 3D. It supports diverse backends, including finite-volume, spectral-element, finite-element, lattice-Boltzmann, and fully differentiable solvers for gradient-enhanced optimization. Across environments, RL agents consistently discover robust control principles, such as boundary-layer manipulation, acoustic-feedback disruption, and wake reorganization, yielding drag reductions exceeding 90% in canonical configurations. Critically, we demonstrate zero-shot transfer: agents trained only on a simplified channel flow achieve 38% friction-drag reduction on an unseen 3D wing section at chord Reynolds number Re=200,000 reducing exploration costs by four orders of magnitude versus direct on-wing optimization. This suggests RL agents uncover essential physics rather than configuration-specific patterns, pointing toward generalizable control. HydroGym offers extensible, scalable community infrastructure for fluid dynamics, machine learning, and control research.

Benchmarking has been the cornerstone of progress in computer vision, natural language processing, and the broader deep learning domain, driving algorithmic innovation through standardized datasets and reproducible evaluation protocols. The growing availability of large-scale Computational Fluid Dynamics (CFD) datasets has opened new opportunities for applying machine learning to aerodynamic and engineering design. Yet, despite this progress, there exists no standardized benchmark for large-scale numerical simulations in engineering design. In this work, we introduce CarBench, the first comprehensive benchmark dedicated to large-scale 3D car aerodynamics, performing a large-scale evaluation of state-of-the-art models on DrivAerNet++, the largest public dataset for automotive aerodynamics, containing over 8,000 high-fidelity car simulations. We assess eleven architectures spanning neural operator methods (e.g., Fourier Neural Operator), geometric deep learning (PointNet, RegDGCNN, PointMAE, PointTransformer), transformer-based neural solvers (Transolver, Transolver++, AB-UPT), and implicit field networks (TripNet). Beyond standard interpolation tasks, we perform cross-category experiments in which transformer-based solvers trained on a single car archetype are evaluated on unseen categories. Our analysis covers predictive accuracy, physical consistency, computational efficiency, and statistical uncertainty. To accelerate progress in data-driven engineering, we open-source the benchmark framework, including training pipelines, uncertainty estimation routines based on bootstrap resampling, and pretrained model weights, establishing the first reproducible foundation for large-scale learning from high-fidelity CFD simulations, available at https://github.com/Mohamedelrefaie/CarBench.