A.1 Data Generation A.1.1 Closure modeling For the present case, the initial condition is given by: u (x, 0) = The 2048 mesh point high-resolution solution is generated using the Fourier-Galerkin spectral method [41] with the 4th order Runge-Kutta method for time stepping. From the box-filtered initial condition, the 32-point low-resolution solution is conducted using central differencing for the spatial derivatives. This choice does not introduce additional artificial viscosity; thus, the solution without closure is naturally unstable. The high-resolution is computed at a small time-step, yet is down-sampled temporally at an interval equal to the low-resolution time step size t =0.0075 s. In this setting, u can be regarded as fully resolved, thus the numerical residual r ( u), defined in Eq. (10), is zero.

Crashworthiness assessment is a critical aspect of automotive design, traditionally relying on high - fidelity finite element (FE) simulations that are computationally expensive and time - consuming. This work presents an exploratory comparative study on devel oping machine learning - based surrogate models for efficient prediction of structural deformation in crash scenarios using the NVIDIA PhysicsNeMo framework. Given the limited prior work applying machine learning to structural crash dynamics, the primary con tribution lies in demonstrating the feasibility and engineering utility of th e various modeling approaches explored in this work . We investigate two state - of - the - art neural network architectures for modeling crash dynamics: MeshGraphNet, a graph neural network that is widely employed in physics - based simulations, and Transolver, a transformer - based architecture with a physics - aware attention mechanism designed to maintain linear computational complexity with respect to geometric scale. Additionally, we examine three strategies for modeling transient dynamics: Time - Conditional, where the tempor al state is directly parameterized by time; the standard Autoregressive approach, which recursively propagates predictions through time; and a stability - enhanced Autoregressive scheme incorporating rollout - based training to improve prediction accuracy and long - term temporal consistency. The models are evaluated on a comprehensive Body - in - White (BIW) crash dataset comprising 150 detailed FE simulations using LS - DYNA . The dataset represents a structurally rich vehicle assembly with over 200 components, including 38 key components featuring variable thickness distributions to capture realistic manufacturing variability. Each model utilizes the undeformed mesh geometry and component characteristics as inputs to predict the spatiotemporal evolution of the deformed mesh during the crash sequence. Evaluation results show that the models capture the overall deformation trends with reasonable fidelity, demonstrating the feasibility of applying machine learning to structural crash dynamics. Although not yet matching full FE accuracy, the models achieve orders - of - magnitude reductions in computational cost, enabling rapid design exploration and early - stage optimization in crashworthiness evaluation. In the modern automotive industry, the assurance of vehicle safety is not merely a design consideration but a fundamental engineering imperative. Crashworthiness, defined as the ability of a vehicle's structure to protect its occupants during an impact, is a primary driver of the design and validation process. This focus is enforced by stringent government regulations and amplified by co nsumer safety rating programs, which have created a competitive landscape where occupant protection is a key market differ entiator. The engineering challenge of crashworthiness is profoundly complex, extending beyond simple structural strength.

A.1 Data Generation A.1.1 Closure modeling For the present case, the initial condition is given by: u (x, 0) = The 2048 mesh point high-resolution solution is generated using the Fourier-Galerkin spectral method [41] with the 4th order Runge-Kutta method for time stepping. From the box-filtered initial condition, the 32-point low-resolution solution is conducted using central differencing for the spatial derivatives. This choice does not introduce additional artificial viscosity; thus, the solution without closure is naturally unstable. The high-resolution is computed at a small time-step, yet is down-sampled temporally at an interval equal to the low-resolution time step size t =0.0075 s. In this setting, u can be regarded as fully resolved, thus the numerical residual r ( u), defined in Eq. (10), is zero.

This works investigates the generalization capabilities of MeshGraphNets (MGN) [Pfaff et al. Learning Mesh-Based Simulation with Graph Networks. ICML 2021] to unseen geometries for fluid dynamics, e.g. predicting the flow around a new obstacle that was not part of the training data. For this purpose, we create a new benchmark dataset for data-driven computational fluid dynamics (CFD) which extends DeepMind's flow around a cylinder dataset by including different shapes and multiple objects. We then use this new dataset to extend the generalization experiments conducted by DeepMind on MGNs by testing how well an MGN can generalize to different shapes. In our numerical tests, we show that MGNs can sometimes generalize well to various shapes by training on a dataset of one obstacle shape and testing on a dataset of another obstacle shape.

Remarkable progress in the development of Deep Learning Weather Prediction (DLWP) models positions them to become competitive with traditional numerical weather prediction (NWP) models. Indeed, a wide number of DLWP architectures -- based on various backbones, including U-Net, Transformer, Graph Neural Network (GNN), and Fourier Neural Operator (FNO) -- have demonstrated their potential at forecasting atmospheric states. However, due to differences in training protocols, forecast horizons, and data choices, it remains unclear which (if any) of these methods and architectures are most suitable for weather forecasting and for future model development. Here, we step back and provide a detailed empirical analysis, under controlled conditions, comparing and contrasting the most prominent DLWP models, along with their backbones. We accomplish this by predicting synthetic two-dimensional incompressible Navier-Stokes and real-world global weather dynamics. In terms of accuracy, memory consumption, and runtime, our results illustrate various tradeoffs. For example, on synthetic data, we observe favorable performance of FNO; and on the real-world WeatherBench dataset, our results demonstrate the suitability of ConvLSTM and SwinTransformer for short-to-mid-ranged forecasts. For long-ranged weather rollouts of up to 365 days, we observe superior stability and physical soundness in architectures that formulate a spherical data representation, i.e., GraphCast and Spherical FNO. In addition, we observe that all of these model backbones ``saturate,'' i.e., none of them exhibit so-called neural scaling, which highlights an important direction for future work on these and related models.

Engineering components must meet increasing technological demands in ever shorter development cycles. To face these challenges, a holistic approach is essential that allows for the concurrent development of part design, material system and manufacturing process. Current approaches employ numerical simulations, which however quickly becomes computation-intensive, especially for iterative optimization. Data-driven machine learning methods can be used to replace time- and resource-intensive numerical simulations. In particular, MeshGraphNets (MGNs) have shown promising results. They enable fast and accurate predictions on unseen mesh geometries while being fully differentiable for optimization. However, these models rely on large amounts of expensive training data, such as numerical simulations. Physics-informed neural networks (PINNs) offer an opportunity to train neural networks with partial differential equations instead of labeled data, but have not been extended yet to handle time-dependent simulations of arbitrary meshes. This work introduces PI-MGNs, a hybrid approach that combines PINNs and MGNs to quickly and accurately solve non-stationary and nonlinear partial differential equations (PDEs) on arbitrary meshes. The method is exemplified for thermal process simulations of unseen parts with inhomogeneous material distribution. Further results show that the model scales well to large and complex meshes, although it is trained on small generic meshes only.

Recent developments in Machine Learning approaches for modelling physical systems have begun to mirror the past development of numerical methods in the computational sciences. In this survey, we begin by providing an example of this with the parallels between the development trajectories of graph neural network acceleration for physical simulations and particle-based approaches. We then give an overview of simulation approaches, which have not yet found their way into state-of-the-art Machine Learning methods and hold the potential to make Machine Learning approaches more accurate and more efficient. We conclude by presenting an outlook on the potential of these approaches for making Machine Learning models for science more efficient.

Reduced-order models based on physics are a popular choice in cardiovascular modeling due to their efficiency, but they may experience reduced accuracy when working with anatomies that contain numerous junctions or pathological conditions. We develop one-dimensional reduced-order models that simulate blood flow dynamics using a graph neural network trained on three-dimensional hemodynamic simulation data. Given the initial condition of the system, the network iteratively predicts the pressure and flow rate at the vessel centerline nodes. Our numerical results demonstrate the accuracy and generalizability of our method in physiological geometries comprising a variety of anatomies and boundary conditions. Our findings demonstrate that our approach can achieve errors below 2% and 3% for pressure and flow rate, respectively, provided there is adequate training data. As a result, our method exhibits superior performance compared to physics-based one-dimensional models, while maintaining high efficiency at inference time.