arXiv subjects

Rajeev K. Jaiman

Publications and source records attributed to Rajeev K. Jaiman.

At least 19 recordsLinked to original sources

arXiv2026

Cavitation Acoustic Perturbation Equations: A Computational Framework for Source-Resolved Multiphase Hydroacoustics

This work develops a cavitation-consistent acoustic perturbation framework for predicting sound generation and propagation in cavitating flows. Unlike conventional acoustic perturbation equations for single-phase or weakly compressible flows, the proposed formulation embeds cavitation physics directly into the acoustic equations. The cavitation acoustic perturbation equations (CAPE) incorporate vapor mass transfer, mixture compressibility, and pressure-rate effects within a unified formulation, allowing cavitation-induced noise sources to be resolved in the computational domain. The numerical framework is verified using one-dimensional wave-propagation problems. The solutions become insensitive to further mesh and time-step refinement, the perfectly matched layer suppresses boundary reflections, and the predicted attenuation over a range of source frequencies follows Stokes' sound attenuation law. The framework is then applied to cavitating flow past a circular cylinder and a NACA hydrofoil. The non-cavitating benchmark shows dipole-like radiation associated with unsteady loading, whereas cavitating cases exhibit monopole-like or geometry-modulated radiation caused by volumetric phase change. Source-term analyses identify tonal frequencies associated with vortex shedding, cavity shedding, and collapse-induced excitation. The phase-change terms provide a direct volumetric contribution to the monopole-like source, while localized collapse events appear through amplification of the pressure-rate source. The proposed framework extends acoustic perturbation methods to cavitating multiphase flows and provides an efficient tool for hydroacoustic prediction, source localization, and mechanism analysis in marine and hydraulic applications.

physics.flu-dyn↗

arXiv2026

A Mesh-Adaptive Hypergraph Neural Network for Unsteady Flow Around Oscillating and Rotating Structures

Graph neural networks, recently introduced into the field of fluid flow surrogate modeling, have been successfully applied to model the temporal evolution of various fluid flow systems. Existing applications, however, are mostly restricted to cases where the domain is time-invariant. The present work extends the application of graph neural network-based modeling to fluid flow around structures rotating with respect to a certain axis. Specifically, we propose to apply a graph neural network-based surrogate model with part of the mesh/graph co-rotating with the structure and part of the mesh/graph static. A single layer of interface cells are constructed at the interface between the two parts and are allowed to distort and adapt, which helps in circumventing the difficulty of interpolating information encoded by the neural network at every graph neural network layer. Dedicated reconstruction and re-projection schemes are designed to counter the error caused by the distortion and connectivity change of the interface cells. The effectiveness of our proposed framework is examined on two test cases: (i) fluid flow around a 2D oscillating airfoil, and (ii) fluid flow past a 3D rotating cube. Our results show that the model achieves stable rollout predictions over hundreds or even a thousand time steps. We further demonstrate that one could enforce accurate, error-bounded prediction results by incorporating the measurements from sparse pressure sensors. In addition to the accurate flow field predictions, the lift and drag force predictions closely match with the computational fluid dynamics calculations, highlighting the potential of the framework for modeling fluid flow around rotating structures, and paving the path towards a graph neural network-based surrogate model for more complex scenarios like flow around marine propellers.

physics.flu-dyn↗

arXiv2025

An Efficient and Accurate Surrogate Modeling of Flapping Dynamics in Inverted Elastic Foils using Hypergraph Neural Networks

Cantilevered elastic foils can undergo self-induced, large-amplitude flapping when subject to fluid flow, a widely observed phenomenon of fluid-structure interaction, from fluttering leaves or the movement of fish fins. When harnessed in steady currents, these oscillations enable the extraction of kinetic energy from the flow. However, accurately predicting these dynamics requires high-fidelity simulations that are prohibitively expensive to perform across the broad configuration space needed for design optimization. To address this, we develop a novel graph neural network (GNN) surrogate for the inverted foil problem, modeled as an elastically mounted rigid foil undergoing trailing-edge pitching in uniform flow. The coupled fluid-structure dynamics are solved using a Petrov-Galerkin finite element method with an arbitrary Lagrangian-Eulerian formulation, providing high-fidelity data for training and validation. The surrogate uses a rotation-equivariant, quasi-monolithic GNN architecture: structural mesh motion is compressed via proper orthogonal decomposition and advanced through a multilayer perceptron. At the same time, the GNN evolves the flow field consistent with system states. Specifically, this study extends the hypergraph framework to flexible, self-oscillating foils, capturing the nonlinear coupling between vortex dynamics and structural motion. The GNN surrogate achieves less than 1.5% error in predicting tip displacement and force coefficients over thousands of time steps, while accurately reproducing dominant vortex-shedding frequencies. The model captures energy transfer metrics within 3% of full-order simulations, demonstrating both accuracy and long-term stability. These results show a new, efficient surrogate for long-horizon prediction of unsteady flow-structure dynamics in energy-harvesting systems.

physics.flu-dyn↗

arXiv2025

HeatGen: A Guided Diffusion Framework for Multiphysics Heat Sink Design Optimization

This study presents a generative optimization framework based on a guided denoising diffusion probabilistic model (DDPM) that leverages surrogate gradients to generate heat sink designs minimizing pressure drop while maintaining surface temperatures below a specified threshold. Geometries are represented using boundary representations of multiple fins, and a multi-fidelity approach is employed to generate training data. Using this dataset, along with vectors representing the boundary representation geometries, we train a denoising diffusion probabilistic model to generate heat sinks with characteristics consistent with those observed in the data. We train two different residual neural networks to predict the pressure drop and surface temperature for each geometry. We use the gradients of these surrogate models with respect to the design variables to guide the geometry generation process toward satisfying the low-pressure and surface temperature constraints. This inference-time guidance directs the generative process toward heat sink designs that not only prevent overheating but also achieve lower pressure drops compared to traditional optimization methods such as CMA-ES. In contrast to traditional black-box optimization approaches, our method is scalable, provided sufficient training data is available. Unlike traditional topology optimization methods, once the model is trained and the heat sink world model is saved, inference under new constraints (e.g., temperature) is computationally inexpensive and does not require retraining. Samples generated using the guided diffusion model achieve pressure drops up to 10 percent lower than the limits obtained by traditional black-box optimization methods. This work represents a step toward building a foundational generative model for electronics cooling.

cs.LG↗

arXiv2025

A Physics-Guided Probabilistic Surrogate Modeling Framework for Digital Twins of Underwater Radiated Noise

Ship traffic is an increasing source of underwater radiated noise in coastal waters, motivating real-time digital twins of ocean acoustics for operational noise mitigation. We present a physics-guided probabilistic framework to predict three-dimensional transmission loss in realistic ocean environments. As a case study, we consider the Salish Sea along shipping routes from the Pacific Ocean to the Port of Vancouver. A dataset of over 30 million source-receiver pairs was generated with a Gaussian beam solver across seasonal sound speed profiles and one-third-octave frequency bands spanning 12.5 Hz to 8 kHz. We first assess sparse variational Gaussian processes (SVGP) and then incorporate physics-based mean functions combining spherical spreading with frequency-dependent absorption. To capture nonlinear effects, we examine deep sigma-point processes and stochastic variational deep kernel learning. The final framework integrates four components: (i) a learnable physics-informed mean that represents dominant propagation trends, (ii) a convolutional encoder for bathymetry along the source-receiver track, (iii) a neural encoder for source, receiver, and frequency coordinates, and (iv) a residual SVGP layer that provides calibrated predictive uncertainty. This probabilistic digital twin facilitates the construction of sound-exposure bounds and worst-case scenarios for received levels. We further demonstrate the application of the framework to ship speed optimization, where predicted transmission loss combined with near-field source models provides sound exposure level estimates for minimizing acoustic impacts on marine mammals. The proposed framework advances uncertainty-aware digital twins for ocean acoustics and illustrates how physics-guided machine learning can support sustainable maritime operations.

cs.LG↗

arXiv2025

A Phase Field Formulation of Frictional Sliding Contact for 3D Fully Eulerian Fluid Structure Interactions

Frictional sliding contact in hydrodynamic environments can be found in a range of engineering applications. Accurate modeling requires an integrated numerical framework capable of resolving large relative motions, multiphase interactions, and nonlinear contact responses. Building on our previously developed fully Eulerian fluid structure formulation, we introduce a phase field based formulation for dynamic frictional contact in 3D. Contact detection is achieved via the overlap of diffuse interfaces of colliding solids. The normal contact response is defined as a volumetric body force proportional to the overlap parameter, while the tangential response is computed using the Coulomb friction model. The direction of the friction forces are derived by projecting phase-averaged relative velocities onto the local tangent plane of colliding bodies. This proposed unified treatment enables the computation of both normal and frictional forces within a single momentum balance equation, avoiding separate velocity fields for individual solids. We present several test cases with increasing complexity to verify and demonstrate our proposed frictional contact model. Verification against the Hertzian contact problem shows excellent agreement with the analytical solution, with errors below $3\%$ in the traction profile. In the sliding block benchmark, the computed displacement profiles closely follow the analytical solution for point-mass systems across multiple friction coefficients. The ironing problem demonstrates stable force predictions under finite deformation, with normal and tangential forces matching kinetic friction laws. The robustness and scalability of the proposed formulation are further demonstrated through a representative ship ice interaction scenario with free surface and frictional sliding effects.

physics.comp-ph↗

arXiv2025

MUTE-DSS: A Digital-Twin-Based Decision Support System for Minimizing Underwater Radiated Noise in Ship Voyage Planning

We present a novel MUTE-DSS, a digital-twin-based decision support system for minimizing underwater radiated noise (URN) during ship voyage planning. It is a ROS2-centric framework that integrates state-of-the-art acoustic models combining a semi-empirical reference spectrum for near-field modeling with 3D ray tracing for propagation losses for far-field modeling, offering real-time computation of the ship noise signature, alongside a data-driven Southern resident killer whale distribution model. The proposed DSS performs a two-stage optimization pipeline: Batch Informed Trees for collision-free ship routing and a genetic algorithm for adaptive ship speed profiling under voyage constraints that minimizes cumulative URN exposure to marine mammals. The effectiveness of MUTE-DSS is demonstrated through case studies of ships operating between the Strait of Georgia and the Strait of Juan de Fuca, comparing optimized voyages against baseline trajectories derived from automatic identification system data. Results show substantial reductions in noise exposure level, up to 7.14 dB, corresponding to approximately an 80.68% reduction in a simplified scenario, and an average 4.90 dB reduction, corresponding to approximately a 67.6% reduction in a more realistic dynamic setting. These results illustrate the adaptability and practical utility of the proposed decision support system.

math.OC↗

arXiv2025

A Reward-Directed Diffusion Framework for Generative Design Optimization

This study presents a generative optimization framework that builds on a fine-tuned diffusion model and reward-directed sampling to generate high-performance engineering designs. The framework adopts a parametric representation of the design geometry and produces new parameter sets corresponding to designs with enhanced performance metrics. A key advantage of the reward-directed approach is its suitability for scenarios in which performance metrics rely on costly engineering simulations or surrogate models (e.g. graph-based, ensemble models, or tree-based) are non-differentiable or prohibitively expensive to differentiate. This work introduces the iterative use of a soft value function within a Markov decision process framework to achieve reward-guided decoding in the diffusion model. By incorporating soft-value guidance during both the training and inference phases, the proposed approach reduces computational and memory costs to achieve high-reward designs, even beyond the training data. Empirical results indicate that this iterative reward-directed method substantially improves the ability of the diffusion models to generate samples with reduced resistance in 3D ship hull design and enhanced hydrodynamic performance in 2D airfoil design tasks. The proposed framework generates samples that extend beyond the training data distribution, resulting in a greater 25 percent reduction in resistance for ship design and over 10 percent improvement in the lift-to-drag ratio for the 2D airfoil design. Successful integration of this model into the engineering design life cycle can enhance both designer productivity and overall design performance.

cs.LG↗

arXiv2025

Predicting Wave Dynamics using Deep Learning with Multistep Integration Inspired Attention and Physics-Based Loss Decomposition

In this paper, we present a physics-based deep learning framework for data-driven prediction of wave propagation in fluid media. The proposed approach, termed Multistep Integration-Inspired Attention (MI2A), combines a denoising-based convolutional autoencoder for reduced latent representation with an attention-based recurrent neural network with long-short-term memory cells for time evolution of reduced coordinates. This proposed architecture draws inspiration from classical linear multistep methods to enhance stability and long-horizon accuracy in latent-time integration. Despite the efficiency of hybrid neural architectures in modeling wave dynamics, autoregressive predictions are often prone to accumulating phase and amplitude errors over time. To mitigate this issue within the MI2A framework, we introduce a novel loss decomposition strategy that explicitly separates the training loss function into distinct phase and amplitude components. We assess the performance of MI2A against two baseline reduced-order models trained with standard mean-squared error loss: a sequence-to-sequence recurrent neural network and a variant using Luong-style attention. To demonstrate the effectiveness of the MI2A model, we consider three benchmark wave propagation problems of increasing complexity, namely one-dimensional linear convection, the nonlinear viscous Burgers equation, and the two-dimensional Saint-Venant shallow water system. Our results demonstrate that the MI2A framework significantly improves the accuracy and stability of long-term predictions, accurately preserving wave amplitude and phase characteristics. Compared to the standard long-short term memory and attention-based models, MI2A-based deep learning exhibits superior generalization and temporal accuracy, making it a promising tool for real-time wave modeling.

cs.LG↗

arXiv2024

A Graph Neural Network Surrogate Model for Multi-Objective Fluid-Acoustic Shape Optimization

This article presents a graph neural network (GNN) based surrogate modeling approach for fluid-acoustic shape optimization. The GNN model transforms mesh-based simulations into a computational graph, enabling global prediction of pressure and velocity flow fields around solid boundaries. We employ signed distance functions to implicitly represent geometries on unstructured nodes represented by the graph neural network. The trained graph neural network is employed here to predict the flow field around various airfoil shapes. The median relative error in the prediction of pressure and velocity for 300 test cases is 1-2\%. The predicted flow field is employed to extract the fluid force coefficients and the velocity profile of the boundary layer. The boundary layer velocity profile is then used to predict the flow field and noise levels, allowing the direct integration of the coupled fluid-acoustic analysis in the shape optimization algorithm. The fluid-acoustic shape optimization is extended to multi-objective shape optimization by minimizing trailing edge noise while maximizing the aerodynamic performance of airfoil surfaces. The results show that the overall sound pressure level of the optimized airfoil decreases by 13.9\% (15.82 dBA), and the lift coefficient increases by 7.2\%, for a fixed set of operating conditions. The proposed GNN-based integrated surrogate modeling with the shape optimization algorithm exhibits a computational speed-up of three orders of magnitude compared to while maintaining reasonable accuracy compared to full-order online optimization applications. The GNN-based surrogate model offers an efficient computational framework for fluid-acoustic shape optimization via adaptive morphing of structures.

physics.flu-dyn↗

arXiv2024

H-SIREN: Improving implicit neural representations with hyperbolic periodic functions

Implicit neural representations (INR) have been recently adopted in various applications ranging from computer vision tasks to physics simulations by solving partial differential equations. Among existing INR-based works, multi-layer perceptrons with sinusoidal activation functions find widespread applications and are also frequently treated as a baseline for the development of better activation functions for INR applications. Recent investigations claim that the use of sinusoidal activation functions could be sub-optimal due to their limited supported frequency set as well as their tendency to generate over-smoothed solutions. We provide a simple solution to mitigate such an issue by changing the activation function at the first layer from $\sin(x)$ to $\sin(\sinh(2x))$. We demonstrate H-SIREN in various computer vision and fluid flow problems, where it surpasses the performance of several state-of-the-art INRs.

cs.CV↗

arXiv2024

An Efficient Phase-field Framework for Contact Dynamics between Deformable Solids in Fluid Flow

Elastic contact in hydrodynamic environments is a complex multiphysics phenomenon and can be found in applications ranging from engineering to biological systems. Understanding the intricacies of this coupled problem requires the development of a generalized framework capable of handling topological changes and transitioning implicitly from FSI conditions to solid-solid contact conditions. We propose a mono-field interface advancing method for handling multibody contact simulations in submerged environments. Given the physical demands of the problem, we adopt a phase-field based fully Eulerian approach to resolve the multiphase and multibody interactions in the system. We employ a stabilized finite element formulation and a partitioned iterative procedure to solve the unified momentum equation comprising solid and fluid dynamics coupled with the Allen-Cahn phase-field equation. We introduce a contact force approach to handle smooth elastic-elastic and elastic-rigid contact based on the overlap of the diffused interfaces of two colliding bodies. We propose a novel approach to extend the model for multibody contact simulations while using a single phase-field function for all the solids. The method is based on updating the solid boundaries at every time step and checking for collisions among them. The developed approach eliminates the need to solve multiple phase field equations and multiple strain equations at every time step. This reduces the overall computational time by nearly $16\%$ compared to a multi phase-field approach. The implemented model is verified for smooth dry contact and FSI contact scenarios. Using the proposed framework, we demonstrate the collision dynamics between multiple bodies submerged in an open liquid tank.

physics.flu-dyn↗

arXiv2024

A multi-objective optimization framework for reducing the impact of ship noise on marine mammals

The underwater radiated noise (URN) emanating from ships presents a significant threat to marine mammals, given their heavy reliance on hearing for essential life activities. The intensity of URN from ships is directly correlated to the speed, making speed reduction a crucial operational mitigation strategy. This paper presents a new multi-objective optimization framework to optimize the ship speed for effective URN mitigation without compromising fuel consumption. The proposed framework addresses a fixed-path voyage scheduling problem, incorporating two objective functions namely (i) noise intensity levels and (ii) fuel consumption. The optimization is performed using the state-of-the-art non-dominated sorting genetic algorithm under voyage constraints. A 2D ocean acoustic environment, comprising randomly scattered marine mammals of diverse audiogram groups and realistic conditions, including sound speed profiles and bathymetry, is simulated. To estimate the objective functions, we consider empirical relations for fuel consumption and near-field noise modeling together with a ray-tracing approach for far-field noise propagation. The optimization problem is solved using the genetic algorithm to determine the Pareto solutions and subsequently the trade-off solution. The effectiveness of the optimization framework is demonstrated via both simplified tests and practical case studies involving a large container ship. A comparative analysis illustrates the adaptability of the optimization framework across different oceanic environments, affirming its potential as a robust tool for reducing the URN from shipping.

math.OC↗

arXiv2024

Wake Interference Effects on Flapping Dynamics of Elastic Inverted Foil

Using high-fidelity simulations, we study the self-induced flapping dynamics of an inverted elastic foil when it is placed in tandem with a stationary circular cylinder. The effect of wake interference on the inverted foil's coupled dynamics is examined at a fixed Reynolds number ($Re$) as a function of non-dimensional bending rigidity ($K_{B}$) and the structure to fluid mass ratio ($m^{*}$). Our results show that there exists a critical $K_{B, Cr} = 0.25$, above which the downstream foil is synchronized with the unsteady wake, and the cylinder controls the flapping response and the wake vortex dynamics. During synchronization, two additional flapping modes namely, small and moderate amplitude flapping mode are observed as a function of decreasing $K_{B}$. Below $K_{B, Cr}$, the downstream foil undergoes self-induced large-amplitude flapping (LAF) similar to an isolated foil counterpart. When the dynamics of the downstream foil are analyzed for a range of $m^{*}$, we can characterize the response dynamics into two regions, namely low and high sensitivity. The high sensitivity region is observed when the dynamics are controlled by the cylinder oscillations, i.e., for foils with high stiffness. In this regime, the foil dynamics negatively correlate to $K_{B}$ and $m^{*}$. The low sensitivity region is observed when the downstream foil is no longer synchronized with the wake and undergoes an LAF response, with dynamics that are weakly correlated to $K_{B}$. A new non-dimensional parameter is proposed that combines the effect of the foil's inertia and elastic forces and can capture the foil's response when it is subjected to wake interference effects. The findings from this study aim to generalize our understanding of the self-induced flapping dynamics of inverted foils in an array configuration and have relevance to the development of inverted foil-based renewable energy harvesters.

physics.flu-dyn↗

arXiv2024

Continual Learning of Range-Dependent Transmission Loss for Underwater Acoustic using Conditional Convolutional Neural Net

There is a significant need for precise and reliable forecasting of the far-field noise emanating from shipping vessels. Conventional full-order models based on the Navier-Stokes equations are unsuitable, and sophisticated model reduction methods may be ineffective for accurately predicting far-field noise in environments with seamounts and significant variations in bathymetry. Recent advances in reduced-order models, particularly those based on convolutional and recurrent neural networks, offer a faster and more accurate alternative. These models use convolutional neural networks to reduce data dimensions effectively. However, current deep-learning models face challenges in predicting wave propagation over long periods and for remote locations, often relying on auto-regressive prediction and lacking far-field bathymetry information. This research aims to improve the accuracy of deep-learning models for predicting underwater radiated noise in far-field scenarios. We propose a novel range-conditional convolutional neural network that incorporates ocean bathymetry data into the input. By integrating this architecture into a continual learning framework, we aim to generalize the model for varying bathymetry worldwide. To demonstrate the effectiveness of our approach, we analyze our model on several test cases and a benchmark scenario involving far-field prediction over Dickin's seamount in the Northeast Pacific. Our proposed architecture effectively captures transmission loss over a range-dependent, varying bathymetry profile. This architecture can be integrated into an adaptive management system for underwater radiated noise, providing real-time end-to-end mapping between near-field ship noise sources and received noise at the marine mammal's location.

cs.LG↗

arXiv2023

A finite element framework for fluid-structure interaction of turbulent cavitating flows with flexible structures

We present a finite element framework for the numerical prediction of cavitating turbulent flows interacting with flexible structures. The vapor-fluid phases are captured through a homogeneous mixture model, with a scalar transport equation governing the spatio-temporal evolution of cavitation dynamics. High-density gradients in the two-phase cavitating flow motivate the use of a positivity-preserving Petrov-Galerkin stabilization method in the variational framework. A mass transfer source term introduces local compressibility effects arising as a consequence of phase change. The turbulent fluid flow is modeled through a dynamic subgrid-scale method for large eddy simulations. The flexible structure is represented by a set of eigenmodes, obtained through the modal analysis of the linear elasticity equations. A partitioned iterative approach is adopted to couple the structural dynamics and cavitating fluid flow, where the deforming flow domain is described by an arbitrary Lagrangian-Eulerian frame of reference. Through the numerical validation study, we establish the fidelity of the proposed framework by comparing it against experimental and numerical studies for both rigid and flexible hydrofoils in cavitating flows. Based on the validation study conducted over a flexible NACA66 rectangular hydrofoil, we elucidate the role of cavity and vortex shedding in governing the structural dynamics of a flexible hydrofoil.

physics.flu-dyn↗

arXiv2023

Predicting fluid-structure interaction with graph neural networks

We present a rotation equivariant, quasi-monolithic graph neural network framework for the reduced-order modeling of fluid-structure interaction systems. With the aid of an arbitrary Lagrangian-Eulerian formulation, the system states are evolved temporally with two sub-networks. The movement of the mesh is reduced to the evolution of several coefficients via complex-valued proper orthogonal decomposition, and the prediction of these coefficients over time is handled by a single multi-layer perceptron. A finite element-inspired hypergraph neural network is employed to predict the evolution of the fluid state based on the state of the whole system. The structural state is implicitly modeled by the movement of the mesh on the solid-fluid interface; hence it makes the proposed framework quasi-monolithic. The effectiveness of the proposed framework is assessed on two prototypical fluid-structure systems, namely the flow around an elastically-mounted cylinder, and the flow around a hyperelastic plate attached to a fixed cylinder. The proposed framework tracks the interface description and provides stable and accurate system state predictions during roll-out for at least 2000 time steps, and even demonstrates some capability in self-correcting erroneous predictions. The proposed framework also enables direct calculation of the lift and drag forces using the predicted fluid and mesh states, in contrast to existing convolution-based architectures. The proposed reduced-order model via graph neural network has implications for the development of physics-based digital twins concerning moving boundaries and fluid-structure interactions.

physics.flu-dyn↗

arXiv2023

A Finite Element-Inspired Hypergraph Neural Network: Application to Fluid Dynamics Simulations

An emerging trend in deep learning research focuses on the applications of graph neural networks (GNNs) for mesh-based continuum mechanics simulations. Most of these learning frameworks operate on graphs wherein each edge connects two nodes. Inspired by the data connectivity in the finite element method, we present a method to construct a hypergraph by connecting the nodes by elements rather than edges. A hypergraph message-passing network is defined on such a node-element hypergraph that mimics the calculation process of local stiffness matrices. We term this method a finite element-inspired hypergraph neural network, in short FEIH($ϕ$)-GNN. We further equip the proposed network with rotation equivariance, and explore its capability for modeling unsteady fluid flow systems. The effectiveness of the network is demonstrated on two common benchmark problems, namely the fluid flow around a circular cylinder and airfoil configurations. Stabilized and accurate temporal roll-out predictions can be obtained using the $ϕ$-GNN framework within the interpolation Reynolds number range. The network is also able to extrapolate moderately towards higher Reynolds number domain out of the training range.

physics.flu-dyn↗