Computational Fluid Dynamics

Also known as CFD

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Overview

Computational fluid dynamics (CFD) uses numerical methods to solve equations describing fluid motion, turbulence, heat transfer, and related processes. Finite-volume and finite-element solvers, alongside RANS, LES, and DNS, provide established ways to simulate flows; neural operators, graph networks, physics-informed neural networks, and generative models are increasingly used as surrogates or for reconstruction. These approaches can reduce the cost of simulation, support design and flow control, and infer fields from sparse measurements. A central challenge is balancing speed with physical consistency and reliable generalization across geometries and regimes, especially over long predictions; learned models still need validation against trusted simulations and experiments.

Latest papers

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279 papers37 papers in the last four weeks, up 9% on the four weeks before. 0.2% of all new papers.

Apr 7, 2026

Topological Characterization of Churn Flow and Unsupervised Correction to the Wu Flow-Regime Map in Small-Diameter Vertical Pipes

Churn flow-the chaotic, oscillatory regime in vertical two-phase flow-has lacked a quantitative mathematical definition for over 4040 years. We introduce the first topology-based characterization using Euler Characteristic Surfaces (ECS). We formulate unsupervised regime discovery as Multiple Kernel Learning (MKL), blending two complementary ECS-derived kernels-temporal alignment (L1L^1 distance on the χ(s,t)χ(s,t) surface) and amplitude statistics (scale-wise mean, standard deviation, max, min)-with gas velocity. Applied to 3737 unlabeled air-water trials from Montana Tech, the self-calibrating framework learns weights βECS=0.14β_{ECS}=0.14, βamp=0.50β_{amp}=0.50, βugs=0.36β_{ugs}=0.36, placing 64%64\% of total weight on topology-derived features (βECS+βampβ_{ECS} + β_{amp}). The ECS-inferred slug/churn transition lies +3.81+3.81 m/s above Wu et al.'s (2017) prediction in 22-in. tubing, quantifying reports that existing models under-predict slug persistence in small-diameter pipes where interfacial tension and wall-to-wall interactions dominate flow. Cross-facility validation on 947947 Texas A&M University images confirms 1.9×1.9\times higher topological complexity in churn vs. slug (p<10−5p < 10^{-5}). Applied to 4545 TAMU pseudo-trials, the same unsupervised framework achieves 95.6%95.6\% 44-class accuracy and 100%100\% churn recall-without any labeled training data-matching or exceeding supervised baselines that require thousands of annotated examples. This work provides the first mathematical definition of churn flow and demonstrates that unsupervised topological descriptors can challenge and correct widely adopted mechanistic models.

Brady Koenig, Sushovan Majhi, Atish Mitra +2

Department of Petroleum Engineering, Montana Technological University, USA · Data Science, George Washington University, USA · Department of Mathematical Sciences, Montana Technological University, USA

Computational Fluid DynamicsTopological Data AnalysisUnsupervised Clustering
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Apr 7, 2026

Asymptotic-Preserving Neural Networks for Viscoelastic Parameter Identification in Multiscale Blood Flow Modeling

Mathematical models and numerical simulations offer a non-invasive way to explore cardiovascular phenomena, providing access to quantities that cannot be measured directly. In this study, we start with a one-dimensional multiscale blood flow model that describes the viscoelastic properties of arterial walls, and we focus on improving its practical applicability by addressing a major challenge: determining, in a reliable way, the viscoelastic parameters that control how arteries deform under pulsatile pressure. To achieve this, we employ Asymptotic-Preserving Neural Networks that embed the governing physical principles of the multiscale viscoelastic blood flow model within the learning procedure. This framework allows us to infer the viscoelastic parameters while simultaneously reconstructing the time-dependent evolution of the state variables of blood vessels. With this approach, pressure waveforms are estimated from readily accessible patient-specific data, i.e., cross-sectional area and velocity measurements from Doppler ultrasound, in vascular segments where direct pressure measurements are not available. Different numerical simulations, conducted in both synthetic and patient-specific scenarios, show the effectiveness of the proposed methodology.

Giulia Bertaglia, Raffaella Fiamma Cabini

Department of Environmental and Prevention Sciences, University of Ferrara, Corso Ercole I d’Este 32, 44121, Ferrara, Italy

Parameter EstimationComputational Fluid Dynamics
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Apr 7, 2026

Optimal-Transport-Guided Functional Flow Matching for Turbulent Field Generation in Hilbert Space

High-fidelity modeling of turbulent flows requires capturing complex spatiotemporal dynamics and multi-scale intermittency, posing a fundamental challenge for traditional knowledge-based systems. While deep generative models, such as diffusion models and Flow Matching, have shown promising performance, they are fundamentally constrained by their discrete, pixel-based nature. This limitation restricts their applicability in turbulence computing, where data inherently exists in a functional form. To address this gap, we propose Functional Optimal Transport Conditional Flow Matching (FOT-CFM), a generative framework defined directly in infinite-dimensional function space. Unlike conventional approaches defined on fixed grids, FOT-CFM treats physical fields as elements of an infinite-dimensional Hilbert space, and learns resolution-invariant generative dynamics directly at the level of probability measures. By integrating Optimal Transport (OT) theory, we construct deterministic, straight-line probability paths between noise and data measures in Hilbert space. This formulation enables simulation-free training and significantly accelerates the sampling process. We rigorously evaluate the proposed system on a diverse suite of chaotic dynamical systems, including the Navier-Stokes equations, Kolmogorov Flow, and Hasegawa-Wakatani equations, all of which exhibit rich multi-scale turbulent structures. Experimental results demonstrate that FOT-CFM achieves superior fidelity in reproducing high-order turbulent statistics and energy spectra compared to state-of-the-art baselines.

Li Kunpeng, Wan Chenguang, Qu Zhisong +5

Flow-Based Generative ModelingFlow MatchingComputational Fluid DynamicsOptimal Transport
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Apr 7, 2026

Multiscale Physics-Informed Neural Network for Complex Fluid Flows with Long-Range Dependencies

Fluid flows are governed by the nonlinear Navier-Stokes equations, which can manifest multiscale dynamics even from predictable initial conditions. Predicting such phenomena remains a formidable challenge in scientific machine learning, particularly regarding convergence speed, data requirements, and solution accuracy. In complex fluid flows, these challenges are exacerbated by long-range spatial dependencies arising from distant boundary conditions, which typically necessitate extensive supervision data to achieve acceptable results. We propose the Domain-Decomposed and Shifted Physics-Informed Neural Network (DDS-PINN), a framework designed to resolve such multiscale interactions with minimal supervision. By utilizing localized networks with a unified global loss, DDS-PINN captures global dependencies while maintaining local precision. The robustness of the approach is demonstrated across a suite of benchmarks, including a multiscale linear differential equation, the nonlinear Burgers' equation, and data-free Navier-Stokes simulations of flat-plate boundary layers. Finally, DDS-PINN is applied to the computationally challenging backward-facing step (BFS) problem; for laminar regimes (Re = 100), the model yields results comparable to computational fluid dynamics (CFD) without the need for any data, accurately predicting boundary layer thickness, separation, and reattachment lengths. For turbulent BFS flow at Re = 10,000, the framework achieves convergence to O(10^-4) using only 500 random supervision points (< 0.3 % of the total domain), outperforming established methods like Residual-based Attention-PINN in accuracy. This approach demonstrates strong potential for the super-resolution of complex turbulent flows from sparse experimental measurements.

Prashant Kumar, Rajesh Ranjan

Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, India 208016

Domain Decomposition MethodsNeural PDE SolversPhysics-Informed MLPDE SolvingScientific MLComputational Fluid Dynamics
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Apr 2, 2026

A Multimodal Vision Transformer-based Modeling Framework for Prediction of Fluid Flows in Energy Systems

Computational fluid dynamics (CFD) simulations of complex fluid flows in energy systems are prohibitively expensive due to strong nonlinearities and multiscale-multiphysics interactions. In this work, we present a transformer-based modeling framework for prediction of fluid flows, and demonstrate it for high-pressure gas injection phenomena relevant to reciprocating engines. The approach employs a hierarchical Vision Transformer (SwinV2-UNet) architecture that processes multimodal flow datasets from multi-fidelity simulations. The model architecture is conditioned on auxiliary tokens explicitly encoding the data modality and time increment. Model performance is assessed on two different tasks: (1) spatiotemporal rollouts, where the model autoregressively predicts the flow state at future times; and (2) feature transformation, where the model infers unobserved fields/views from observed fields/views. We train separate models on multimodal datasets generated from in-house CFD simulations of argon jet injection into a nitrogen environment, encompassing multiple grid resolutions, turbulence models, and equations of state. The resulting data-driven models learn to generalize across resolutions and modalities, accurately forecasting the flow evolution and reconstructing missing flow-field information from limited views. This work demonstrates how large vision transformer-based models can be adapted to advance predictive modeling of complex fluid flow systems.

Kiran Yalamanchi, Shivam Barwey, Ibrahim Jarrah +1

Transportation and Power Systems Division, Argonne National Laboratory, Lemont, IL, USA

Computational Fluid DynamicsPDE Surrogate ModelingMultifidelity Modeling
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Apr 2, 2026

Application of parametric Shallow Recurrent Decoder Network to magnetohydrodynamic flows in liquid metal blankets of fusion reactors

Magnetohydrodynamic (MHD) phenomena play a pivotal role in the design and operation of nuclear fusion systems, where electrically conducting fluids (such as liquid metals or molten salts employed in reactor blankets) interact with magnetic fields of varying intensity and orientation, influencing the resulting flow dynamics. The numerical solution of MHD models entails the resolution of highly nonlinear, multiphysics systems of equations, which can become computationally demanding, particularly in multi-query, parametric, or real-time contexts. This study investigates a fully data-driven framework for MHD state reconstruction that integrates dimensionality reduction through Singular Value Decomposition (SVD) with the SHallow REcurrent Decoder (SHRED), a neural network architecture designed to reconstruct the full spatio-temporal state from sparse time-series measurements of selected observables, including previously unseen parametric configurations. The SHRED methodology is applied to a three-dimensional geometry representative of a portion of a WCLL blanket cell, in which lead-lithium flows around a water-cooled tube. Multiple magnetic field configurations are examined, including constant toroidal fields, combined toroidal-poloidal fields, and time-dependent magnetic fields. Across all considered scenarios, SHRED achieves high reconstruction accuracy, robustness, and generalization to magnetic field intensities, orientations, and temporal evolutions not seen during training. Notably, in the presence of time-varying magnetic fields, the model accurately infers the temporal evolution of the magnetic field itself using temperature measurements alone. Overall, the findings identify SHRED as a computationally efficient, data-driven, and flexible approach for MHD state reconstruction, with significant potential for real-time monitoring, diagnostics and control in fusion reactor systems.

M. Lo Verso, C. Introini, E. Cervi +3

Department of Energy, Politecnico di Milano, Milano, 20133, Italy · MATHEP Group, Dept. of Energy "Galileo Ferraris", Politecnico di Torino, Torino, Italy · Autodesk Research, 6 Agar Street, London UK +1

MagnetohydrodynamicsComputational Fluid DynamicsNeural Surrogate Modeling
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Mar 31, 2026

Autonomous Adaptive Solver Selection for Chemistry Integration via Reinforcement Learning

The computational cost of stiff chemical kinetics remains a dominant bottleneck in reacting-flow simulation, yet hybrid integration strategies are typically driven by hand-tuned heuristics or supervised predictors that make myopic decisions from instantaneous local state. We introduce a constrained reinforcement learning (RL) framework that autonomously selects between an implicit BDF integrator (CVODE) and a quasi-steady-state (QSS) solver during chemistry integration. Solver selection is cast as a Markov decision process. The agent learns trajectory-aware policies that account for how present solver choices influence downstream error accumulation, while minimizing computational cost under a user-prescribed accuracy tolerance enforced through a Lagrangian reward with online multiplier adaptation. Across sampled 0D homogeneous reactor conditions, the RL-adaptive policy achieves a mean speedup of approximately 3×3\times, with speedups ranging from 1.11×1.11\times to 10.58×10.58\times, while maintaining accurate ignition delays and species profiles for a 106-species n-dodecane mechanism and adding approximately 1%1\% inference overhead. Without retraining, the 0D-trained policy transfers to 1D counterflow diffusion flames over strain rates 1010--2000 s−12000~\mathrm{s}^{-1}, delivering consistent ≈2.2×\approx 2.2\times speedup relative to CVODE while preserving near-reference temperature accuracy and selecting CVODE at only 1212--15%15\% of space-time points. Overall, the results demonstrate the potential of the proposed reinforcement learning framework to learn problem-specific integration strategies while respecting accuracy constraints, thereby opening a pathway toward adaptive, self-optimizing workflows for multiphysics systems with spatially heterogeneous stiffness.

Eloghosa Ikponmwoba, Opeoluwa Owoyele

Department of Mechanical and Industrial Engineering Louisiana State University Baton Rouge, LA 70803

Reinforcement LearningConstrained RLComputational Fluid Dynamics
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Mar 31, 2026

LGFNet: Local-Global Fusion Network with Fidelity Gap Delta Learning for Multi-Source Aerodynamics

The precise fusion of computational fluid dynamic (CFD) data, wind tunnel tests data, and flight tests data in aerodynamic area is essential for obtaining comprehensive knowledge of both localized flow structures and global aerodynamic trends across the entire flight envelope. However, existing methodologies often struggle to balance high-resolution local fidelity with wide-range global dependency, leading to either a loss of sharp discontinuities or an inability to capture long-range topological correlations. We propose Local-Global Fusion Network (LGFNet) for multi-scale feature decomposition to extract this dual-natured aerodynamic knowledge. To this end, LGFNet combines a spatial perception layer that integrates a sliding window mechanism with a relational reasoning layer based on self-attention, simultaneously reinforcing the continuity of fine-grained local features (e.g., shock waves) and capturing long-range flow information. Furthermore, the fidelity gap delta learning (FGDL) strategy is proposed to treat CFD data as a "low-frequency carrier" to explicitly approximate nonlinear discrepancies. This approach prevents unphysical smoothing while inheriting the foundational physical trends from the simulation baseline. Experiments demonstrate that LGFNet achieves state-of-the-art (SOTA) performance in both accuracy and uncertainty reduction across diverse aerodynamic scenarios.

Qinye Zhu, Yu Xiang, Jun Zhang +1

School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, China

Multi-Scale Feature FusionMultifidelity ModelingComputational Fluid Dynamics
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Mar 31, 2026

Lie Generator Networks for Nonlinear Partial Differential Equations

Linear dynamical systems are fully characterized by their eigenspectra, accessible directly from the generator of the dynamics. For nonlinear systems governed by partial differential equations, no equivalent theory exists. We introduce Lie Generator Network-Koopman (LGN-KM), a neural operator that lifts nonlinear dynamics into a linear latent space and learns the continuous-time Koopman generator (LkL_k) through a decomposition Lk=S−DkL_k = S - D_k, where SS is skew-symmetric representing conservative inter-modal coupling, and DkD_k is a positive-definite diagonal encoding modal dissipation. This architectural decomposition enforces stability and enables interpretability through direct spectral access to the learned dynamics. On two-dimensional Navier--Stokes turbulence, the generator recovers the known dissipation scaling and a complete multi-branch dispersion relation from trajectory data alone with no physics supervision. Independently trained models at different flow regimes recover matched gauge-invariant spectral structure, exposing a gauge freedom in the Koopman lifting. Because the generator is provably stable, it enables guaranteed long-horizon stability, continuous-time evaluation at arbitrary time, and physics-informed cross-viscosity model transfer.

Shafayeth Jamil, Rehan Kapadia

Ming Hsieh Department of Electrical and Computer Engineering University of Southern California

Koopman Operator LearningPDE Operator LearningNonlinear System IdentificationComputational Fluid Dynamics
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Mar 30, 2026

Data-informed lifting line theory

We present a data-driven framework that extends the predictive capability of classical lifting-line theory (LLT) to a wider aerodynamic regime by incorporating higher-fidelity aerodynamic data from panel method simulations. A neural network architecture with a convolutional layer followed by fully connected layers is developed, comprising two parallel subnetworks to separately process spanwise collocation points and global geometric/aerodynamic inputs such as angle of attack, chord, twist, airfoil distribution, and sweep. Among several configurations tested, this architecture is most effective in learning corrections to LLT outputs. The trained model captures higher-order three-dimensional effects in spanwise lift and drag distributions in regimes where LLT is inaccurate, such as low aspect ratios and high sweep, and generalizes well to wing configurations outside both the LLT regime and the training data range. The method retains LLT's computational efficiency, enabling integration into aerodynamic optimization loops and early-stage aircraft design studies. This approach offers a practical path for embedding high-fidelity corrections into low-order methods and may be extended to other aerodynamic prediction tasks, such as propeller performance.

Arjun Sharma, Jonas A. Actor, Peter A. Bosler

Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, 87185, USA

Computational Fluid DynamicsNeural Surrogate Modeling
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Mar 30, 2026

FluidFlow: a flow-matching generative model for fluid dynamics surrogates on unstructured meshes

Computational fluid dynamics (CFD) provides high-fidelity simulations of fluid flows but remains computationally expensive for many-query applications. In recent years deep learning (DL) has been used to construct data-driven fluid-dynamic surrogate models. In this work we consider a different learning paradigm and embrace generative modelling as a framework for constructing scalable fluid-dynamics surrogate models. We introduce FluidFlow, a generative model based on conditional flow-matching, a recent alternative to diffusion models that learns deterministic transport maps between noise and data distributions. FluidFlow is specifically designed to operate directly on CFD data defined on both structured and unstructured meshes alike, without the needs to perform any mesh interpolation pre-processing and preserving geometric fidelity. We assess the capabilities of FluidFlow using two different core neural network architectures, a U-Net and diffusion transformer (DiT), and condition their learning on physically meaningful parameters. The methodology is validated on two benchmark problems of increasing complexity: prediction of pressure coefficients along an airfoil boundary across different operating conditions, and prediction of pressure and friction coefficients over a full three-dimensional aircraft geometry discretized on a large unstructured mesh. In both cases, FluidFlow outperform strong multilayer perceptron baselines, achieving significantly lower error metrics and improved generalisation across operating conditions. Notably, the transformer-based architecture enables scalable learning on large unstructured datasets while maintaining high predictive accuracy. These results demonstrate that flow-matching generative models provide an effective and flexible framework for surrogate modelling in fluid dynamics, with potential for realistic engineering and scientific applications.

David Ramos, Lucas Lacasa, Fermín Gutiérrez +2

ETSIAE-UPM-School of Aeronautics, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros 3, E-28040 Madrid, Spain · Institute for Cross-Disciplinary Physics and Complex Systems (IFISC, CSIC-UIB), 07122 Palma de Mallorca (Spain) · Center for Computational Simulation, Universidad Politécnica de Madrid, Campus de Montegancedo, Boadilla del Monte, 28660 Madrid, Spain

PDE Surrogate ModelingComputational Fluid DynamicsConditional Generative Modeling
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Mar 30, 2026

Koopman-based surrogate modeling for reinforcement-learning-control of Rayleigh-Benard convection

Training reinforcement learning (RL) agents to control fluid dynamics systems is computationally expensive due to the high cost of direct numerical simulations (DNS) of the governing equations. Surrogate models offer a promising alternative by approximating the dynamics at a fraction of the computational cost, but their feasibility as training environments for RL is limited by distribution shifts, as policies induce state distributions not covered by the surrogate training data. In this work, we investigate the use of Linear Recurrent Autoencoder Networks (LRANs) for accelerating RL-based control of 2D Rayleigh-Bénard convection. We evaluate two training strategies: a surrogate trained on precomputed data generated with random actions, and a policy-aware surrogate trained iteratively using data collected from an evolving policy. Our results show that while surrogate-only training leads to reduced control performance, combining surrogates with DNS in a pretraining scheme recovers state-of-the-art performance while reducing training time by more than 40%. We demonstrate that policy-aware training mitigates the effects of distribution shift, enabling more accurate predictions in policy-relevant regions of the state space.

Tim Plotzki, Sebastian Peitz

TU Dortmund University & Lamarr Institute for Machine Learning and Artificial Intelligence, Dortmund, Germany

RL ControlPDE Surrogate ModelingDistribution ShiftComputational Fluid Dynamics
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Mar 30, 2026

SIMR-NO: A Spectrally-Informed Multi-Resolution Neural Operator for Turbulent Flow Super-Resolution

Reconstructing high-resolution turbulent flow fields from severely under-resolved observations is a fundamental inverse problem in computational fluid dynamics and scientific machine learning. Classical interpolation methods fail to recover missing fine-scale structures, while existing deep learning approaches rely on convolutional architectures that lack the spectral and multiscale inductive biases necessary for physically faithful reconstruction at large upscaling factors. We introduce the Spectrally-Informed Multi-Resolution Neural Operator (SIMR-NO), a hierarchical operator learning framework that factorizes the ill-posed inverse mapping across intermediate spatial resolutions, combines deterministic interpolation priors with spectrally gated Fourier residual corrections at each stage, and incorporates local refinement modules to recover fine-scale spatial features beyond the truncated Fourier basis. The proposed method is evaluated on Kolmogorov-forced two-dimensional turbulence, where 128×128128\times128 vorticity fields are reconstructed from extremely coarse 8×88\times8 observations representing a 16×16\times downsampling factor. Across 201 independent test realizations, SIMR-NO achieves a mean relative ℓ2\ell_2 error of 26.04%26.04\% with the lowest error variance among all methods, reducing reconstruction error by 31.7%31.7\% over FNO, 26.0%26.0\% over EDSR, and 9.3%9.3\% over LapSRN. Beyond pointwise accuracy, SIMR-NO is the only method that faithfully reproduces the ground-truth energy and enstrophy spectra across the full resolved wavenumber range, demonstrating physically consistent super-resolution of turbulent flow fields.

Muhammad Abid, Omer San

Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville, TN 37996, USA.

Neural OperatorsComputational Fluid DynamicsFourier Neural Operator
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Mar 28, 2026

Sparse Autoencoders as a Steering Basis for Phase Synchronization in Graph-Based CFD Surrogates

Graph-based surrogate models provide fast alternatives to high-fidelity CFD solvers, but their opaque latent spaces and limited controllability restrict use in safety-critical settings. A key failure mode in oscillatory flows is phase drift, where predictions remain qualitatively correct but gradually lose temporal alignment with observations, limiting use in digital twins and closed-loop control. Correcting this through retraining is expensive and impractical during deployment. We ask whether phase drift can instead be corrected post hoc by manipulating the latent space of a frozen surrogate. We propose a phase-steering framework for pretrained graph-based CFD models that combines the right representation with the right intervention mechanism. To obtain disentangled representation for effective steering, we use sparse autoencoders (SAEs) on frozen MeshGraphNet embeddings. To steer dynamics, we move beyond static per-feature interventions such as scaling or clamping, and introduce a temporally coherent, phase-aware method. Specifically, we identify oscillatory feature pairs with Hilbert analysis, project spatial fields into low-rank temporal coefficients via SVD, and apply smooth time-varying rotations to advance or delay periodic modes while preserving amplitude-phase structure. Using a representation-agnostic setup, we compare SAE-based steering with PCA and raw embedding spaces under the same intervention pipeline. Results show that sparse, disentangled representations outperform dense or entangled ones, while static interventions fail in this dynamical setting. Overall, this work shows that latent-space steering can be extended from semantic domains to time-dependent physical systems when interventions respect the underlying dynamics, and that the same sparse features used for interpretability can also serve as physically meaningful control axes.

Yeping Hu, Ruben Glatt, Shusen Liu

Computational Engineering Division Lawrence Livermore National Laboratory Livermore, CA 94550 · Center for Applied Scientific Computing Lawrence Livermore National Laboratory Livermore, CA 94550

Computational Fluid DynamicsPDE Surrogate ModelingLatent Space Optimization
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Mar 26, 2026

Anchored-Branched Steady-state WInd Flow Transformer (AB-SWIFT): a metamodel for 3D atmospheric flow in urban environments

Air flow modeling at a local scale is essential for applications such as pollutant dispersion modeling or wind farm modeling. To circumvent costly Computational Fluid Dynamics (CFD) computations, deep learning surrogate models have recently emerged as promising alternatives. However, in the context of urban air flow, deep learning models struggle to adapt to the high variations of the urban geometry and to large mesh sizes. To tackle these challenges, we introduce Anchored Branched Steady-state WInd Flow Transformer (AB-SWIFT), a transformer-based model with an internal branched structure uniquely designed for atmospheric flow modeling. We train our model on a specially designed database of atmospheric simulations around randomised urban geometries and with a mixture of unstable, neutral, and stable atmospheric stratifications. Our model reaches the best accuracy on all predicted fields compared to state-of-the-art transformers and graph-based models. Our code and data is available at https://github.com/cerea-daml/abswift.

Armand de Villeroché, Rem-Sophia Mouradi, Vincent Le Guen +5

CEREA, ENPC, EDF R&D, Institut Polytechnique de Paris, Île-de-France, France · EDF R&D, Île-de-France, France · SINCLAIR AI Laboratory, Saclay, Île-de-France, France +1

Neural Surrogate ModelingPDE Surrogate ModelingTransformerComputational Fluid Dynamics
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Mar 26, 2026

DSO: Dual-Scale Neural Operators for Stable Long-term Fluid Dynamics Forecasting

Long-term fluid dynamics forecasting is a critically important problem in science and engineering. While neural operators have emerged as a promising paradigm for modeling systems governed by partial differential equations (PDEs), they often struggle with long-term stability and precision. We identify two fundamental failure modes in existing architectures: (1) local detail blurring, where fine-scale structures such as vortex cores and sharp gradients are progressively smoothed, and (2) global trend deviation, where the overall motion trajectory drifts from the ground truth during extended rollouts. We argue that these failures arise because existing neural operators treat local and global information processing uniformly, despite their inherently different evolution characteristics in physical systems. To bridge this gap, we propose the Dual-Scale Neural Operator (DSO), which explicitly decouples information processing into two complementary modules: depthwise separable convolutions for fine-grained local feature extraction and an MLP-Mixer for long-range global aggregation. Through numerical experiments on vortex dynamics, we demonstrate that nearby perturbations primarily affect local vortex structure while distant perturbations influence global motion trends, providing empirical validation for our design choice. Extensive experiments on turbulent flow benchmarks show that DSO achieves state-of-the-art accuracy while maintaining robust long-term stability, reducing prediction error by over 88% compared to existing neural operators.

Huanshuo Dong, Hao Wu, Hong Wang +2

University of Science and Technology of China, Hefei, China · Tsinghua University, Beijing, China · Institute of Automation, Chinese Academy of Sciences, Beijing, China +1

PDE Operator LearningNeural OperatorsComputational Fluid DynamicsLong-Term Time Series Forecasting
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Mar 25, 2026

Project and Generate: Divergence-Free Neural Operators for Incompressible Flows

Learning-based models for fluid dynamics often operate in unconstrained function spaces, leading to physically inadmissible, unstable simulations. While penalty-based methods offer soft regularization, they provide no structural guarantees, resulting in spurious divergence and long-term collapse. In this work, we introduce a unified framework that enforces the incompressible continuity equation as a hard, intrinsic constraint for both deterministic and generative modeling. First, to project deterministic models onto the divergence-free subspace, we integrate a differentiable spectral Leray projection grounded in the Helmholtz-Hodge decomposition, which restricts the regression hypothesis space to physically admissible velocity fields. Second, to generate physically consistent distributions, we show that simply projecting model outputs is insufficient when the prior is incompatible. To address this, we construct a divergence-free Gaussian reference measure via a curl-based pushforward, ensuring the entire probability flow remains subspace-consistent by construction. Experiments on 2D Navier-Stokes equations demonstrate exact incompressibility up to discretization error and substantially improved stability and physical consistency.

Xigui Li, Hongwei Zhang, Ruoxi Jiang +6

Artificial Intelligence Innovation and Incubation Institute, Fudan University, Shanghai, China · Shanghai Academy of Artificial Intelligence for Science, Shanghai, China

Neural OperatorsComputational Fluid DynamicsPhysics-Informed Generative ModelingPhysics-Informed Neural Operators
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Mar 24, 2026

Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations

This work establishes rigorous first-of-its-kind upper bounds on the generalization error for the method of approximating solutions to the (d+1)-dimensional incompressible Navier-Stokes equations by training depth-2 neural networks trained via the unsupervised Physics-Informed Neural Network (PINN) framework. This is achieved by bounding the Rademacher complexity of the PINN risk. For appropriately weight bounded net classes our derived generalization bounds do not explicitly depend on the network width and our framework characterizes the generalization gap in terms of the fluid's kinematic viscosity and loss regularization parameters. In particular, the resulting sample complexity bounds are dimension-independent. Our generalization bounds suggest using novel activation functions for solving fluid dynamics. We provide empirical validation of the suggested activation functions and the corresponding bounds on a PINN setup solving the Taylor-Green vortex benchmark.

Sebastien Andre-Sloan, Dibyakanti Kumar, Alejandro F Frangi +1

Department of Computer Science, The University of Manchester

Generalization BoundsPhysics-Informed MLComputational Fluid DynamicsStatistical Learning Theory
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Mar 22, 2026

Pretrained Video Models as Differentiable Physics Simulators for Urban Wind Flows

Designing urban spaces that provide pedestrian wind comfort and safety requires time-resolved Computational Fluid Dynamics (CFD) simulations, but their current computational cost makes extensive design exploration impractical. We introduce WinDiNet (Wind Diffusion Network), a pretrained video diffusion model that is repurposed as a fast, differentiable surrogate for this task. Starting from LTX-Video, a 2B-parameter latent video transformer, we fine-tune on 10,000 2D incompressible CFD simulations over procedurally generated building layouts. A systematic study of training regimes, conditioning mechanisms, and VAE adaptation strategies, including a physics-informed decoder loss, identifies a configuration that outperforms purpose-built neural PDE solvers. The resulting model generates full 112-frame rollouts in under a second. As the surrogate is end-to-end differentiable, it doubles as a physics simulator for gradient-based inverse optimization: given an urban footprint layout, we optimize building positions directly through backpropagation to improve wind safety as well as pedestrian wind comfort. Experiments on single- and multi-inlet layouts show that the optimizer discovers effective layouts even under challenging multi-objective configurations, with all improvements confirmed by ground-truth CFD simulations.

Janne Perini, Rafael Bischof, Moab Arar +4

1Computational Design Lab, ETH Zurich, Switzerland · 2Tel Aviv University, Israel · 3Architecture and Building Systems, ETH Zurich, Switzerland +2

Video Diffusion ModelsPDE Surrogate ModelingDiffusion Model Fine-TuningDifferentiable PhysicsComputational Fluid Dynamics
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Mar 20, 2026

Modeling subgrid scale production rates on complex meshes using graph neural networks

Large-eddy simulations (LES) require closures for filtered production rates because the resolved fields do not contain all correlations that govern chemical source terms. We develop a graph neural network (GNN) that predicts filtered species production rates on non-uniform meshes from inputs of filtered mass fractions and temperature. Direct numerical simulations of turbulent premixed hydrogen-methane jet flames with hydrogen fractions of 10%, 50%, and 80% provide the dataset. All fields are Favre filtered with the filter width matched to the operating mesh, and learning is performed on subdomain graphs constructed from mesh-point connectivity. A compact set of reactants, intermediates, and products is used, and their filtered production rates form the targets. The model is trained on 10% and 80% blends and evaluated on the unseen 50% blend to test cross-composition generalization. The GNN is compared against an unclosed reference that evaluates rates at the filtered state, and a convolutional neural network baseline that requires remeshing. Across in-distribution and out-of-distribution cases, the GNN yields lower errors and closer statistical agreement with the reference data. Furthermore, the model demonstrates robust generalization across varying filter widths without retraining, maintaining bounded errors at coarser spatial resolutions. A backward facing step configuration further confirms prediction efficacy on a practically relevant geometry. These results highlight the capability of GNNs as robust data-driven closure models for LES on complex meshes.

Priyabrat Dash, Mathis Bode, Konduri Aditya

Department of Computational and Data Sciences, Indian Institute of Science, Bengaluru, 560012, KA, India · Jülich Supercomputing Centre, Forschungszentrum Jülich GmbH, Jülich, 52425, NRW, Germany

Computational Fluid DynamicsGraph Neural Networks
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Mar 20, 2026

An Adaptive Machine Learning Framework for Fluid Flow in Dual-Network Porous Media

Porous materials -- natural or engineered -- often exhibit dual pore-network structures that govern processes such as mineral exploration and hydrocarbon recovery from tight shales. Double porosity/permeability (DPP) mathematical models describe incompressible fluid flow through two interacting pore networks with inter-network mass exchange. Despite significant advances in numerical methods, there remains a need for computational frameworks that enable rapid forecasting, data assimilation, and reliable inverse analysis. To address this, we present a physics-informed neural network (PINN) framework for forward and inverse modeling of DPP systems. The proposed approach encodes the governing equations in mixed form, along with boundary conditions, directly into the loss function, with adaptive weighting strategies to balance their contributions. Key features of the framework include adaptive weight tuning, dynamic collocation point selection, and the use of shared trunk neural architectures to efficiently capture the coupled behavior of the dual pore networks. It is inherently mesh-free, making it well-suited for complex geometries typical of porous media. It accurately captures discontinuities in solution fields across layered domains without introducing spurious oscillations commonly observed in classical finite element formulations. Importantly, the framework is well-suited for inverse analysis, enabling robust parameter identification in scenarios where key physical quantities -- such as the mass transfer coefficient in DPP models -- are difficult to measure directly. In addition, a systematic convergence analysis is provided to rigorously assess the stability, accuracy, and reliability of the method. The effectiveness and computational advantages of the approach are demonstrated through a series of representative numerical experiments.

V. S. Maduri, K. B. Nakshatrala

Department of Civil & Environmental Engineering University of Houston, Houston, Texas 77204, USA.

Neural PDE SolversPorous Media FlowPhysics-Informed MLPDE Inverse ProblemsComputational Fluid Dynamics
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Mar 19, 2026

Model Order Reduction of Cerebrovascular Hemodynamics Using POD_Galerkin and Reservoir Computing_based Approach

We investigate model order reduction (MOR) strategies for simulating unsteady hemodynamics within cerebrovascular systems, contrasting a physics-based intrusive approach with a data-driven non-intrusive framework. High-fidelity 3D Computational Fluid Dynamics (CFD) snapshots of an idealised basilar artery bifurcation are first compressed into a low-dimensional latent space using Proper Orthogonal Decomposition (POD). We evaluate the performance of a POD-Galerkin (POD-G) model, which projects the Navier-Stokes equations onto the reduced basis, against a POD-Reservoir Computing (POD-RC) model that learns the temporal evolution of coefficients through a recurrent architecture. A multi-harmonic and multi-amplitude training signal is introduced to improve training efficiency. Both methodologies achieve computational speed-ups on the order of 10^2 to 10^3 compared to full-order simulations, demonstrating their potential as efficient and accurate surrogates for predicting flow quantities such as wall shear stress.

Rahul Halder, Arash Hajisharifi, Kabir Bakhshaei +1

Mathematics Area, mathLab, SISSA, via Bonomea 265, I-34136 Trieste, Italy · Biorobotics Institute, SMART Lab, Sant’Anna School of Advanced Studies, Pisa, Italy · Department of Computer Science, University of Pisa, Pisa, Italy

Reduced-Order ModelingPDE Surrogate ModelingComputational Fluid DynamicsReservoir ComputingPDE Solving
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Mar 17, 2026

Physics-integrated neural differentiable modeling for immersed boundary systems

Accurately, efficiently, and stably computing complex fluid flows and their evolution near solid boundaries over long horizons remains challenging. Conventional numerical solvers require fine grids and small time steps to resolve near-wall dynamics, resulting in high computational costs, while purely data-driven surrogate models accumulate rollout errors and lack robustness under extrapolative conditions. To address these issues, this study extends existing neural PDE solvers by developing a physics-integrated differentiable framework for long-horizon prediction of immersed-boundary flows. A key design aspect of the framework includes an important improvement, namely the structural integration of physical principles into an end-to-end differentiable architecture incorporating a PDE-based intermediate velocity module and a multi-direct forcing immersed boundary module, both adhering to the pressure-projection procedure for incompressible flow computation. The computationally expensive pressure projection step is substituted with a learned implicit correction using ConvResNet blocks to reduce cost, and a sub-iteration strategy is introduced to separate the embedded physics module's stability requirement from the surrogate model's time step, enabling stable coarse-grid autoregressive rollouts with large effective time increments. The framework uses only single-step supervision for training, eliminating long-horizon backpropagation and reducing training time to under one hour on a single GPU. Evaluations on benchmark cases of flow past a stationary cylinder and a rotationally oscillating cylinder at Re=100 show the proposed model consistently outperforms purely data-driven, physics-loss-constrained, and coarse-grid numerical baselines in flow-field fidelity and long-horizon stability, while achieving an approximately 200-fold inference speedup over the high-resolution solver.

Chenglin Li, Hang Xu, Jianting Chen +1

State Key Lab of Ocean Engineering, School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China · State Key Laboratory of Maritime Technology and Safety, Shanghai Ship and Shipping Research Institute Co., Ltd, Shanghai 200135, China.

PDE Surrogate ModelingComputational Fluid DynamicsNeural PDE SolversDifferentiable Physics
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Mar 16, 2026

A convolutional autoencoder and neural ODE surrogate modeling framework applied to transient counterflow flames

A novel convolutional autoencoder and neural ODE (CAE-NODE) framework is proposed for a reduced-order model (ROM) applied to transient 2D counterflow flames, as an extension of AE-NODE methods in homogeneous reactive systems to spatially resolved flows. The multidimensional thermochemical fields (256 x 256 grid, 21 variables) obtained from direct numerical simulations (DNS) are used in training the CAE, where convolutional layers learned the underlying spatial correlations, allowing the CAE to construct an unsupervised 3D latent manifold that is physically meaningful, smooth, and continuous in time. This results in a compression ratio of over 400,000 times. The NODE then subsequently learns the continuous-time dynamics on the latent manifold, enabling the prediction of the full temporal evolution of the flames by integrating forward in time from an initial condition. The results demonstrate that the CAE-NODE can accurately capture the entire transient process, including ignition, flame propagation, and the gradual transition to a non-premixed condition, with excellent agreement with the DNS, while adhering to conservation principles at virtually no computational cost compared to the reference DNS. Predictions remain accurate at strain rates outside the training range. Moreover, despite being unsupervised, the learned latent manifold is highly correlated with the flame-state descriptors such as the progress variable, mixture fraction, and the scalar dissipation rate. This study, for the first time, highlights the potential of CAE-NODE for surrogate modeling of unsteady dynamics of multi-dimensional reacting flows.

Mert Yakup Baykan, Weitao Liu, Mohammad Rafi Malik +4

Department of Aerospace Engineering, KAIST, Daejeon, Republic of Korea · Clean Energy Research Platform (CERP), KAUST, Thuwal, Saudi Arabia · Institute for Reactive Flows (IRST), University of Stuttgart, Pfaffenwaldring 31, 70569 Stuttgart, Germany

AutoencodersPDE Surrogate ModelingConvolutional AutoencoderReduced-Order ModelingNeural ODEsComputational Fluid Dynamics
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Mar 14, 2026

Hierarchy of extreme-event predictability in turbulence revealed by machine learning

Extreme-event predictability in turbulence is strongly state dependent, yet event-by-event predictability horizons are difficult to quantify without access to governing equations or costly perturbation ensembles. Here we train an autoregressive conditional diffusion model on direct numerical simulations of the two-dimensional Kolmogorov flow and use a CRPS-based skill score to define an event-wise predictability horizon. Enstrophy extremes exhibit a pronounced hierarchy: forecast skill persists from ≈1\approx 1 to >4> 4 Lyapunov times across events. Spectral filtering shows that these horizons are controlled predominantly by large-scale structures. Extremes are preceded by intense strain cores organizing quadrupolar vortex packets, whose lifetime sharply separates long- from short-horizon events. These results identify coherent-structure persistence as a governing mechanism for the predictability of turbulence extremes and provide a data-driven route to diagnose predictability limits from observations.

Yuxuan Yang, Chenyu Dong, Gianmarco Mengaldo

National University of Singapore Singapore

Chaotic Time-Series PredictionComputational Fluid DynamicsProbabilistic Forecasting
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Mar 12, 2026

Learning Pore-scale Multiphase Flow from 4D Velocimetry

Multiphase flow in porous media underpins subsurface energy and environmental technologies, including geological CO2_2 storage and underground hydrogen storage, yet pore-scale dynamics in realistic three-dimensional materials remain difficult to characterize and predict. Here we introduce a multimodal learning framework that infers multiphase pore-scale flow directly from time-resolved four-dimensional (4D) micro-velocimetry measurements. The model couples a graph network simulator for Lagrangian tracer-particle motion with a 3D U-Net for voxelized interface evolution. The imaged pore geometry serves as a boundary constraint to the flow velocity and the multiphase interface predictions, which are coupled and updated iteratively at each time step. Trained autoregressively on experimental sequences in capillary-dominated conditions (Ca≈10−6Ca\approx10^{-6}), the learned surrogate captures transient, nonlocal flow perturbations and abrupt interface rearrangements (Haines jumps) over rollouts spanning seconds of physical time, while reducing hour-to-day--scale direct numerical simulations to seconds of inference. By providing rapid, experimentally informed predictions, the framework opens a route to ''digital experiments'' to replicate pore-scale physics observed in multiphase flow experiments, offering an efficient tool for exploring injection conditions and pore-geometry effects relevant to subsurface carbon and hydrogen storage.

Chunyang Wang, Linqi Zhu, Yuxuan Gu +9

Department of Earth Science and Engineering, Imperial College London. · Department of Geology, Ghent University. · Department of Energy Science and Engineering, Stanford University. +2

Porous Media FlowComputational Fluid DynamicsNeural Surrogate ModelingMultimodal Learning
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Mar 11, 2026

A Machine Learning-Enhanced Hopf-Cole Formulation for Nonlinear Gas Flow in Porous Media

Accurate modeling of gas flow through porous media is critical for many technological applications, including reservoir performance prediction, carbon capture and sequestration, and fuel cells and batteries. However, such modeling remains challenging due to strong nonlinear behavior and uncertainty in model parameters. In particular, gas slippage effects described by the Klinkenberg model introduce pressure-dependent permeability, which complicates numerical simulation and obscures deviations from classical Darcy flow behavior. To address these challenges, we present an integrated modeling framework for gas transport in porous media that combines a Klinkenberg-enhanced constitutive relation, Hopf-Cole-transformed mixed-form linear governing equations, a shared-trunk neural network architecture, and a Deep Least-Squares (DeepLS) solver. The Hopf-Cole transformation reformulates the original nonlinear flow equations into an equivalent linear system closely related to the Darcy model, while the mixed formulation, together with a shared-trunk neural architecture, enables simultaneous and accurate prediction of both pressure and velocity fields. A rigorous convergence analysis is performed both theoretically and numerically, establishing the stability and convergence properties of the proposed solver. Importantly, the proposed framework also naturally facilitates inverse modeling of pressure-dependent permeability and slippage parameters from limited or indirect observations, enabling efficient estimation of flow properties that are difficult to measure experimentally. Numerical results demonstrate accurate recovery of flow dynamics and parameters across a wide range of pressure regimes, highlighting the framework's robustness, accuracy, and computational efficiency for gas transport modeling and inversion in tight formations.

V. S. Maduri, K. B. Nakshatrala

Graduate Student, Department of Civil & Environmental Engineering University of Houston, Houston, Texas 77204. · Department of Civil & Environmental Engineering University of Houston, Houston, Texas 77204.

Porous Media FlowComputational Fluid DynamicsNeural PDE SolversPDE Inverse Problems
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Mar 11, 2026

Surrogate models for nuclear fusion with parametric Shallow Recurrent Decoder Networks: applications to magnetohydrodynamics

Magnetohydrodynamic (MHD) effects play a key role in the design and operation of nuclear fusion systems, where electrically conducting fluids (such as liquid metals or molten salts in reactor blankets) interact with magnetic fields of varying intensity and orientation, which affect the resulting flow. The numerical resolution of MHD models involves highly nonlinear multiphysics systems of equations and can become computationally expensive, particularly in multi-query, parametric, or real-time contexts. This work investigates a fully data-driven framework for MHD state reconstruction that combines dimensionality reduction via Singular Value Decomposition (SVD) with the SHallow REcurrent Decoder (SHRED), a neural network architecture designed to recover the full spatio-temporal state from sparse time-series measurements of a limited number of observables. The methodology is applied to a parametric MHD test case involving compressible lead-lithium flow in a stepped channel subjected to thermal gradients and magnetic fields spanning a broad range of intensities. To improve efficiency, the full-order dataset is first compressed using SVD, yielding a reduced representation used as reference truth for training. Only temperature measurements from three sensors are provided as input, while the network reconstructs the full fields of velocity, pressure, and temperature. To assess robustness with respect to sensor placement, thirty randomly generated sensor configurations are tested in ensemble mode. Results show that SHRED accurately reconstructs the full MHD state even for magnetic field intensities not included in the training set. These findings demonstrate the potential of SHRED as a computationally efficient surrogate modeling strategy for fusion-relevant multiphysics problems, enabling low-cost state estimation with possible applications in real-time monitoring and control.

M. Lo Verso, C. Introini, E. Cervi +3

Department of Energy, Politecnico di Milano, Milano, 20133, Italy · MATHEP Group, Dept. of Energy "Galileo Ferraris", Politecnico di Torino, Torino, Italy · Autodesk Research, 6 Agar Street, London UK +1

MagnetohydrodynamicsNeural Surrogate ModelingReduced-Order ModelingComputational Fluid DynamicsRecurrent Neural Networks
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Mar 11, 2026

UniPINN: A Unified PINN Framework for Multi-task Learning of Diverse Navier-Stokes Equations

Physics-Informed Neural Networks (PINNs) have shown promise in solving incompressible Navier-Stokes equations, yet existing approaches are predominantly designed for single-flow settings. When extended to multi-flow scenarios, these methods face three key challenges: (1) difficulty in simultaneously capturing both shared physical principles and flow-specific characteristics, (2) susceptibility to inter-task negative transfer that degrades prediction accuracy, and (3) unstable training dynamics caused by disparate loss magnitudes across heterogeneous flow regimes. To address these limitations, we propose UniPINN, a unified multi-flow PINN framework that integrates three complementary components: a shared-specialized architecture that disentangles universal physical laws from flow-specific features, a cross-flow attention mechanism that selectively reinforces relevant patterns while suppressing task-irrelevant interference, and a dynamic weight allocation strategy that adaptively balances loss contributions to stabilize multi-objective optimization. Extensive experiments on three canonical flows demonstrate that UniPINN effectively unifies multi-flow learning, achieving superior prediction accuracy and balanced performance across heterogeneous regimes while successfully mitigating negative transfer. The source code of this paper will be released on https://github.com/Event-AHU/OpenFusion

Dengdi Sun, Jie Chen, Xiao Wang +1

Key Laboratory of Intelligent Computing & Signal Processing (ICSP), Ministry of Education, School of Artificial Intelligence, Anhui University, Hefei 230601, China · Anhui Provincial Key Laboratory of Multimodal Cognitive Computing, School of Computer Science and Technology, Anhui University, Hefei 230601, China

Physics-Informed MLMulti-Task LearningComputational Fluid DynamicsMulti-Task Optimization
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Mar 10, 2026

Flow Field Reconstruction via Voronoi-Enhanced Physics-Informed Neural Networks with End-to-End Sensor Placement Optimization

(short version abstract, full in article)High-fidelity flow field reconstruction is important in fluid dynamics, but it is challenged by sparse and spatiotemporally incomplete sensor measurements, as well as failures of pre-deployed measurement points that can invalidate pre-trained reconstruction models. Physics-informed neural networks (PINNs) alleviate dependence on large labeled datasets by incorporating governing physics, yet sensor placement optimization, a key factor in reconstruction accuracy and robustness, remains underexplored. In this study, we propose a PINN with Voronoi-enhanced Sensor Optimization (VSOPINN). VSOPINN enables differentiable soft Voronoi construction for sparse sensor data rasterization, end-to-end fusion of centroidal Voronoi tessellation (CVT) with PINNs for adaptive sensor placement, and unified layout optimization for multi-condition flow reconstruction through a shared encoder-multi-decoder architecture. We validate VSOPINN on three representative problems: lid-driven cavity flow, vascular flow, and annular rotating flow. Results show that VSOPINN significantly improves reconstruction accuracy across different Reynolds numbers, adaptively learns effective sensor layouts, and remains robust under partial sensor failure. The study clarifies the intrinsic relationship between sensor placement and reconstruction precision in PINN-based flow field reconstruction.

Renjie Xiao, Bingteng Sun, Yiling Chen +3

Advanced Gas Turbine Laboratory, Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing, 100190, China · National Key Laboratory of Science and Technology on Advanced Light-duty Gas-turbine, Beijing, 100190, China · School of Computer Science and Technology Shandong University Qingdao Shandong 266237 China +3

Computational Fluid DynamicsPhysics-Informed ML
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