Learning Flexible Body Collision Dynamics with Hierarchical Contact Mesh Transformer

Recently, many mesh-based graph neural network (GNN) models have been proposed for modeling complex high-dimensional physical systems. Remarkable achievements have been made in significantly reducing the solving time compared to traditional numerical solvers. These methods are typically designed to i) reduce the computational cost in solving physical dynamics and/or ii) propose techniques to enhance the solution accuracy in fluid and rigid body dynamics. However, it remains under-explored whether they are effective in addressing the challenges of flexible body dynamics, where instantaneous collisions occur within a very short timeframe. In this paper, we present Hierarchical Contact Mesh Transformer (HCMT), which uses hierarchical mesh structures and can learn long-range dependencies (occurred by collisions) among spatially distant positions of a body -- two close positions in a higher-level mesh corresponds to two distant positions in a lower-level mesh. HCMT enables long-range interactions, and the hierarchical mesh structure quickly propagates collision effects to faraway positions. To this end, it consists of a contact mesh Transformer and a hierarchical mesh Transformer (CMT and HMT, respectively). Lastly, we propose a flexible body dynamics dataset, consisting of trajectories that reflect experimental settings frequently used in the display industry for product designs. We also compare the performance of several baselines using well-known benchmark datasets. Our results show that HCMT provides significant performance improvements over existing methods.

Operator-learning-inspired Modeling of Neural Ordinary Differential Equations

Neural ordinary differential equations (NODEs), one of the most influential works of the differential equation-based deep learning, are to continuously generalize residual networks and opened a new field. They are currently utilized for various downstream tasks, e.g., image classification, time series classification, image generation, etc. Its key part is how to model the time-derivative of the hidden state, denoted dh(t)/dt. People have habitually used conventional neural network architectures, e.g., fully-connected layers followed by non-linear activations. In this paper, however, we present a neural operator-based method to define the time-derivative term. Neural operators were initially proposed to model the differential operator of partial differential equations (PDEs). Since the time-derivative of NODEs can be understood as a special type of the differential operator, our proposed method, called branched Fourier neural operator (BFNO), makes sense. In our experiments with general downstream tasks, our method significantly outperforms existing methods.

Hypernetwork-based Meta-Learning for Low-Rank Physics-Informed Neural Networks

In various engineering and applied science applications, repetitive numerical simulations of partial differential equations (PDEs) for varying input parameters are often required (e.g., aircraft shape optimization over many design parameters) and solvers are required to perform rapid execution. In this study, we suggest a path that potentially opens up a possibility for physics-informed neural networks (PINNs), emerging deep-learning-based solvers, to be considered as one such solver. Although PINNs have pioneered a proper integration of deep-learning and scientific computing, they require repetitive time-consuming training of neural networks, which is not suitable for many-query scenarios. To address this issue, we propose a lightweight low-rank PINNs containing only hundreds of model parameters and an associated hypernetwork-based meta-learning algorithm, which allows efficient approximation of solutions of PDEs for varying ranges of PDE input parameters. Moreover, we show that the proposed method is effective in overcoming a challenging issue, known as "failure modes" of PINNs.