Xiaodong Feng, Haojiong Shangguan, Tao Tang, Xiaoliang Wan
原文摘要
In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based formulation where the neural network is defined on a spatiotemporal domain, our methodology utilizes finite element basis functions in the time direction where the space-dependent coefficients are defined as the output of a neural network. We then apply the Galerkin or collocation projection in the time direction to obtain a system of PDEs for the space-dependent coefficients which is approximated in the framework of PINN. The advantages of such a hybrid formulation are twofold: statistical errors are avoided for the integral in the time direction, and the neural network's output can be regarded as a set of reduced spatial basis functions. To further alleviate the difficulties from high dimensionality and low regularity, we have developed an adaptive sampling strategy that refines the training set. More specifically, we use an explicit density model to approximate the distribution induced by the PDE residual and then augment the training set with new time-dependent random samples given by the learned density model. The effectiveness and efficiency of our proposed method have been demonstrated through a series of numerical experiments.
NVIDIA, :, Anis Ahmad, Jon Allen, Malik Aqeel Anwar, Jesse Archer, Maciej Bala, Piotr Barejko 等 90 位作者
原文摘要
We present a shared framework that adapts Cosmos world foundation models for real-time interactive world simulation in Physical AI. By streaming visual observations in response to actions and scene controls, the framework enables users and embodied agents to interact with evolving virtual environments. It supports both predicting the visual consequences of actions and generating observations from physical state maintained by an external simulator. We demonstrate this approach across camera navigation and robot manipulation, controllable video transfer, and multi-view autonomous vehicle (AV) simulation. These examples provide a shared foundation for developing interactive virtual environments for Physical AI training and evaluation. Our code and model checkpoints are available at https://github.com/nvidia/cosmos and https://huggingface.co/collections/nvidia/cosmos3-examples. The project website is available at https://research.nvidia.com/labs/cosmos-lab/cosmos-simulation.