In scenarios with limited available data, training the function-to-function neural PDE solver in an unsupervised manner is essential. However, the efficiency and accuracy of existing methods are constrained by the properties of numerical algorithms, such as finite difference and pseudo-spectral methods, integrated during the training stage. These methods necessitate careful spatiotemporal discretization to achieve reasonable accuracy, leading to significant computational challenges and inaccurate simulations, particularly in cases with substantial spatiotemporal variations. To address these limitations, we propose the Monte Carlo Neural PDE Solver (MCNP Solver) for training unsupervised neural solvers via the PDEs' probabilistic representation, which regards macroscopic phenomena as ensembles of random particles. Compared to other unsupervised methods, MCNP Solver naturally inherits the advantages of the Monte Carlo method, which is robust against spatiotemporal variations and can tolerate coarse step size. In simulating the trajectories of particles, we employ Heun's method for the convection process and calculate the expectation via the probability density function of neighbouring grid points during the diffusion process. These techniques enhance accuracy and circumvent the computational issues associated with Monte Carlo sampling. Our numerical experiments on convection-diffusion, Allen-Cahn, and Navier-Stokes equations demonstrate significant improvements in accuracy and efficiency compared to other unsupervised baselines.

Physics-Informed Neural Network (PINN) has become a commonly used machine learning approach to solve partial differential equations (PDE). But, facing high-dimensional secondorder PDE problems, PINN will suffer from severe scalability issues since its loss includes second-order derivatives, the computational cost of which will grow along with the dimension during stacked back-propagation. In this work, we develop a novel approach that can significantly accelerate the training of Physics-Informed Neural Networks. In particular, we parameterize the PDE solution by the Gaussian smoothed model and show that, derived from Stein’s Identity, the second-order derivatives can be efficiently calculated without back-propagation. We further discuss the model capacity and provide variance reduction methods to address key limitations in the derivative estimation. Experimental results show that our proposed method can achieve competitive error compared to standard PINN training but is significantly faster.

Neural networks have shown great potential in accelerating the solution of partial differential equations (PDEs). Recently, there has been a growing interest in introducing physics constraints into training neural PDE solvers to reduce the use of costly data and improve the generalization ability. However, these physics constraints, based on certain finite dimensional approximations over the function space, must resolve the smallest scaled physics to ensure the accuracy and stability of the simulation, resulting in high computational costs from large input, output, and neural networks. This paper proposes a general acceleration methodology called NeuralStagger by spatially and temporally decomposing the original learning tasks into several coarser-resolution subtasks. We define a coarse-resolution neural solver for each subtask, which requires fewer computational resources, and jointly train them with the vanilla physics-constrained loss by simply arranging their outputs to reconstruct the original solution. Due to the perfect parallelism between them, the solution is achieved as fast as a coarse-resolution neural solver. In addition, the trained solvers bring the flexibility of simulating with multiple levels of resolution. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations, which leads to an additional $10\sim100\times$ speed-up. Moreover, the experiment also shows that the learned model could be well used for optimal control.

Identification and tracking of multiple objects are essential in many applications. As a key enabler of automatic ID technology, RFID has got widespread adoption with item-level tagging in everyday life. However, restricted to the computation capability of passive RFID systems, locating or tracking tags has always been a challenging task. Meanwhile, as a fundamental problem in the field of computer vision, object tracking in images has progressed to a remarkable state especially with the rapid development of deep learning in the past few years. To enable lightweight tracking of a specific target, researchers try to complement computer vision to existing RFID architecture and achieves fine granularity. However, such solution requires calibration of the cameras extrinsic parameters at each new setup, which is not convenient for usage. In this work, we propose Tagview, a pervasive identifying and tracking system that can work in various settings without repetitive calibration efforts. It addresses the challenge by skillfully deploying the RFID antenna and video camera at the identical position and devising a multi-target recognition schema with only the image-level trajectory information. We have implemented Tagview with commercial RFID and camera devices and evaluated it extensively. Experimental results show that our method can archive high accuracy and robustness.