Neural Information Processing Systems http://nips.cc/

We then introduce our method, which builds on and adapts the work of Conrad et al. (2016) and Teymur et al. (2016), and provide a

Probabilistic numerics casts numerical tasks, such the numerical solution of differential equations, as inference problems to be solved.

We then introduce our method, which builds on and adapts the work of Conrad et al. (2016) and Teymur et al. (2016), and provide a

Diffusion models have become emerging generative models. Their sampling process involves multiple steps, and in each step the models predict the noise from a noisy sample. When the models make prediction, the output deviates from the ground truth, and we call such a deviation as \textit{prediction error}. The prediction error accumulates over the sampling process and deteriorates generation quality. This paper introduces a novel technique for statistically measuring the prediction error and proposes the Variance-Reduction Guidance (VRG) method to mitigate this error. VRG does not require model fine-tuning or modification. Given a predefined sampling trajectory, it searches for a new trajectory which has the same number of sampling steps but produces higher quality results. VRG is applicable to both conditional and unconditional generation. Experiments on various datasets and baselines demonstrate that VRG can significantly improve the generation quality of diffusion models. Source code is available at https://github.com/shifengxu/VRG.

This paper presents intersection of Physics informed neural networks (PINNs) and Lie symmetry group to enhance the accuracy and efficiency of solving partial differential equation (PDEs). Various methods have been developed to solve these equations. A Lie group is an efficient method that can lead to exact solutions for the PDEs that possessing Lie Symmetry. Leveraging the concept of infinitesimal generators from Lie symmetry group in a novel manner within PINN leads to significant improvements in solution of PDEs. In this study three distinct cases are discussed, each showing progressive improvements achieved through Lie symmetry modifications and adaptive techniques. State-of-the-art numerical methods are adopted for comparing the progressive PINN models. Numerical experiments demonstrate the key role of Lie symmetry in enhancing PINNs performance, emphasizing the importance of integrating abstract mathematical concepts into deep learning for addressing complex scientific problems adequately.

Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic numerical methods that instead return a Gauss-Markov process defining a probability distribution over the ODE solution. In contrast to prior work, we construct this family such that posterior means match the outputs of the Runge-Kutta family exactly, thus inheriting their proven good properties. Remaining degrees of freedom not identified by the match to Runge-Kutta are chosen such that the posterior probability measure fits the observed structure of the ODE. Our results shed light on the structure of Runge-Kutta solvers from a new direction, provide a richer, probabilistic output, have low computational cost, and raise new research questions.

Probabilistic numerics casts numerical tasks, such the numerical solution of differential equations, as inference problems to be solved.

An internet network service provider manages its network with multiple objectives, such as high quality of service (QoS) and minimum computing resource usage. To achieve these objectives, a reinforcement learning-based (RL) algorithm has been proposed to train its network management agent. Usually, their algorithms optimize their agents with respect to a single static reward formulation consisting of multiple objectives with fixed importance factors, which we call preferences. However, in practice, the preference could vary according to network status, external concerns and so on. For example, when a server shuts down and it can cause other servers' traffic overloads leading to additional shutdowns, it is plausible to reduce the preference of QoS while increasing the preference of minimum computing resource usages. In this paper, we propose new RL-based network management agents that can select actions based on both states and preferences. With our proposed approach, we expect a single agent to generalize on various states and preferences. Furthermore, we propose a numerical method that can estimate the distribution of preference that is advantageous for unbiased training. Our experiment results show that the RL agents trained based on our proposed approach significantly generalize better with various preferences than the previous RL approaches, which assume static preference during training. Moreover, we demonstrate several analyses that show the advantages of our numerical estimation method.

Modal synthesis methods are a long-standing approach for modelling distributed musical systems. In some cases extensions are possible in order to handle geometric nonlinearities. One such case is the high-amplitude vibration of a string, where geometric nonlinear effects lead to perceptually important effects including pitch glides and a dependence of brightness on striking amplitude. A modal decomposition leads to a coupled nonlinear system of ordinary differential equations. Recent work in applied machine learning approaches (in particular neural ordinary differential equations) has been used to model lumped dynamic systems such as electronic circuits automatically from data. In this work, we examine how modal decomposition can be combined with neural ordinary differential equations for modelling distributed musical systems. The proposed model leverages the analytical solution for linear vibration of system's modes and employs a neural network to account for nonlinear dynamic behaviour. Physical parameters of a system remain easily accessible after the training without the need for a parameter encoder in the network architecture. As an initial proof of concept, we generate synthetic data for a nonlinear transverse string and show that the model can be trained to reproduce the nonlinear dynamics of the system. Sound examples are presented.