Estimating Committor Functions via Deep Adaptive Sampling on Rare Transition Paths

The committor functions are central to investigating rare but important events in molecular simulations. It is known that computing the committor function suffers from the curse of dimensionality. Recently, using neural networks to estimate the committor function has gained attention due to its potential for high-dimensional problems. Training neural networks to approximate the committor function needs to sample transition data from straightforward simulations of rare events, which is very inefficient. The scarcity of transition data makes it challenging to approximate the committor function. To address this problem, we propose an efficient framework to generate data points in the transition state region that helps train neural networks to approximate the committor function. We design a Deep Adaptive Sampling method for TRansition paths (DASTR), where deep generative models are employed to generate samples to capture the information of transitions effectively. In particular, we treat a non-negative function in the integrand of the loss functional as an unnormalized probability density function and approximate it with the deep generative model. The new samples from the deep generative model are located in the transition state region and fewer samples are located in the other region. This distribution provides effective samples for approximating the committor function and significantly improves the accuracy. We demonstrate the effectiveness of the proposed method through both simulations and realistic examples.

BAFNet: Bilateral Attention Fusion Network for Lightweight Semantic Segmentation of Urban Remote Sensing Images

Large-scale semantic segmentation networks often achieve high performance, while their application can be challenging when faced with limited sample sizes and computational resources. In scenarios with restricted network size and computational complexity, models encounter significant challenges in capturing long-range dependencies and recovering detailed information in images. We propose a lightweight bilateral semantic segmentation network called bilateral attention fusion network (BAFNet) to efficiently segment high-resolution urban remote sensing images. The model consists of two paths, namely dependency path and remote-local path. The dependency path utilizes large kernel attention to acquire long-range dependencies in the image. Besides, multi-scale local attention and efficient remote attention are designed to construct remote-local path. Finally, a feature aggregation module is designed to effectively utilize the different features of the two paths. Our proposed method was tested on public high-resolution urban remote sensing datasets Vaihingen and Potsdam, with mIoU reaching 83.20% and 86.53%, respectively. As a lightweight semantic segmentation model, BAFNet not only outperforms advanced lightweight models in accuracy but also demonstrates comparable performance to non-lightweight state-of-the-art methods on two datasets, despite a tenfold variance in floating-point operations and a fifteenfold difference in network parameters.

Deep adaptive sampling for surrogate modeling without labeled data

Surrogate modeling is of great practical significance for parametric differential equation systems. In contrast to classical numerical methods, using physics-informed deep learning methods to construct simulators for such systems is a promising direction due to its potential to handle high dimensionality, which requires minimizing a loss over a training set of random samples. However, the random samples introduce statistical errors, which may become the dominant errors for the approximation of low-regularity and high-dimensional problems. In this work, we present a deep adaptive sampling method for surrogate modeling ($\text{DAS}^2$), where we generalize the deep adaptive sampling (DAS) method [62] [Tang, Wan and Yang, 2023] to build surrogate models for low-regularity parametric differential equations. In the parametric setting, the residual loss function can be regarded as an unnormalized probability density function (PDF) of the spatial and parametric variables. This PDF is approximated by a deep generative model, from which new samples are generated and added to the training set. Since the new samples match the residual-induced distribution, the refined training set can further reduce the statistical error in the current approximate solution. We demonstrate the effectiveness of $\text{DAS}^2$ with a series of numerical experiments, including the parametric lid-driven 2D cavity flow problem with a continuous range of Reynolds numbers from 100 to 1000.

A novel approach of solving the CNF-SAT problem

In this paper, we discussed CNF-SAT problem (NP-Complete problem) and analysis two solutions that can solve the problem, the PL-Resolution algorithm and the WalkSAT algorithm. PL-Resolution is a sound and complete algorithm that can be used to determine satisfiability and unsatisfiability with certainty. WalkSAT can determine satisfiability if it finds a model, but it cannot guarantee to find a model even there exists one. However, WalkSAT is much faster than PL-Resolution, which makes WalkSAT more practical; and we have analysis the performance between these two algorithms, and the performance of WalkSAT is acceptable if the problem is not so hard.