Recently, graph neural networks have been gaining a lot of attention to simulate dynamical systems due to their inductive nature leading to zero-shot generalizability. Similarly, physics-informed inductive biases in deep-learning frameworks have been shown to give superior performance in learning the dynamics of physical systems. There is a growing volume of literature that attempts to combine these two approaches. Here, we evaluate the performance of thirteen different graph neural networks, namely, Hamiltonian and Lagrangian graph neural networks, graph neural ODE, and their variants with explicit constraints and different architectures. We briefly explain the theoretical formulation highlighting the similarities and differences in the inductive biases and graph architecture of these systems. Then, we evaluate them on spring, pendulum, and gravitational and 3D deformable solid systems to compare the performance in terms of rollout error, conserved quantities such as energy and momentum, and generalizability to unseen system sizes. Our study demonstrates that GNNs with additional inductive biases, such as explicit constraints and decoupling of kinetic and potential energies, exhibit significantly enhanced performance. Further, all the physics-informed GNNs exhibit zero-shot generalizability to system sizes an order of magnitude larger than the training system, thus providing a promising route to simulate large-scale realistic systems.

Recently, graph neural networks have been gaining a lot of attention to simulate dynamical systems due to their inductive nature leading to zero-shot generalizability. Similarly, physics-informed inductive biases in deep-learning frameworks have been shown to give superior performance in learning the dynamics of physical systems. There is a growing volume of literature that attempts to combine these two approaches. Here, we evaluate the performance of thirteen different graph neural networks, namely, Hamiltonian and Lagrangian graph neural networks, graph neural ODE, and their variants with explicit constraints and different architectures. We briefly explain the theoretical formulation highlighting the similarities and differences in the inductive biases and graph architecture of these systems.

Hemodynamic quantities are valuable biomedical risk factors for cardiovascular pathology such as atherosclerosis. Non-invasive, in-vivo measurement of these quantities can only be performed using a select number of modalities that are not widely available, such as 4D flow magnetic resonance imaging (MRI). In this work, we create a surrogate model for hemodynamic flow field estimation, powered by machine learning. We train graph neural networks that include priors about the underlying symmetries and physics, limiting the amount of data required for training. This allows us to train the model using moderately-sized, in-vivo 4D flow MRI datasets, instead of large in-silico datasets obtained by computational fluid dynamics (CFD), as is the current standard. We create an efficient, equivariant neural network by combining the popular PointNet++ architecture with group-steerable layers. To incorporate the physics-informed priors, we derive an efficient discretisation scheme for the involved differential operators. We perform extensive experiments in carotid arteries and show that our model can accurately estimate low-noise hemodynamic flow fields in the carotid artery. Moreover, we show how the learned relation between geometry and hemodynamic quantities transfers to 3D vascular models obtained using a different imaging modality than the training data. This shows that physics-informed graph neural networks can be trained using 4D flow MRI data to estimate blood flow in unseen carotid artery geometries.

Solving partial differential equations (PDEs) is an important research means in the fields of physics, biology, and chemistry. As an approximate alternative to numerical methods, PINN has received extensive attention and played an important role in many fields. However, PINN uses a fully connected network as its model, which has limited fitting ability and limited extrapolation ability in both time and space. In this paper, we propose PhyGNNet for solving partial differential equations on the basics of a graph neural network which consists of encoder, processer, and decoder blocks. In particular, we divide the computing area into regular grids, define partial differential operators on the grids, then construct pde loss for the network to optimize to build PhyGNNet model. What's more, we conduct comparative experiments on Burgers equation and heat equation to validate our approach, the results show that our method has better fit ability and extrapolation ability both in time and spatial areas compared with PINN.