On the Convergence of Adam under Non-uniform Smoothness: Separability from SGDM and Beyond

This paper aims to clearly distinguish between Stochastic Gradient Descent with Momentum (SGDM) and Adam in terms of their convergence rates. We demonstrate that Adam achieves a faster convergence compared to SGDM under the condition of non-uniformly bounded smoothness. Our findings reveal that: (1) in deterministic environments, Adam can attain the known lower bound for the convergence rate of deterministic first-order optimizers, whereas the convergence rate of Gradient Descent with Momentum (GDM) has higher order dependence on the initial function value; (2) in stochastic setting, Adam's convergence rate upper bound matches the lower bounds of stochastic first-order optimizers, considering both the initial function value and the final error, whereas there are instances where SGDM fails to converge with any learning rate. These insights distinctly differentiate Adam and SGDM regarding their convergence rates. Additionally, by introducing a novel stopping-time based technique, we further prove that if we consider the minimum gradient norm during iterations, the corresponding convergence rate can match the lower bounds across all problem hyperparameters. The technique can also help proving that Adam with a specific hyperparameter scheduler is parameter-agnostic, which hence can be of independent interest.

Deciphering and integrating invariants for neural operator learning with various physical mechanisms

Neural operators have been explored as surrogate models for simulating physical systems to overcome the limitations of traditional partial differential equation (PDE) solvers. However, most existing operator learning methods assume that the data originate from a single physical mechanism, limiting their applicability and performance in more realistic scenarios. To this end, we propose Physical Invariant Attention Neural Operator (PIANO) to decipher and integrate the physical invariants (PI) for operator learning from the PDE series with various physical mechanisms. PIANO employs self-supervised learning to extract physical knowledge and attention mechanisms to integrate them into dynamic convolutional layers. Compared to existing techniques, PIANO can reduce the relative error by 13.6\%-82.2\% on PDE forecasting tasks across varying coefficients, forces, or boundary conditions. Additionally, varied downstream tasks reveal that the PI embeddings deciphered by PIANO align well with the underlying invariants in the PDE systems, verifying the physical significance of PIANO. The source code will be publicly available at: https://github.com/optray/PIANO.

Power-law Dynamic arising from machine learning

We study a kind of new SDE that was arisen from the research on optimization in machine learning, we call it power-law dynamic because its stationary distribution cannot have sub-Gaussian tail and obeys power-law. We prove that the power-law dynamic is ergodic with unique stationary distribution, provided the learning rate is small enough. We investigate its first exist time. In particular, we compare the exit times of the (continuous) power-law dynamic and its discretization. The comparison can help guide machine learning algorithm.

NeuralStagger: accelerating physics-constrained neural PDE solver with spatial-temporal decomposition

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.

Monte Carlo Neural Operator for Learning PDEs via Probabilistic Representation

Neural operators, which use deep neural networks to approximate the solution mappings of partial differential equation (PDE) systems, are emerging as a new paradigm for PDE simulation. The neural operators could be trained in supervised or unsupervised ways, i.e., by using the generated data or the PDE information. The unsupervised training approach is essential when data generation is costly or the data is less qualified (e.g., insufficient and noisy). However, its performance and efficiency have plenty of room for improvement. To this end, we design a new loss function based on the Feynman-Kac formula and call the developed neural operator Monte-Carlo Neural Operator (MCNO), which can allow larger temporal steps and efficiently handle fractional diffusion operators. Our analyses show that MCNO has advantages in handling complex spatial conditions and larger temporal steps compared with other unsupervised methods. Furthermore, MCNO is more robust with the perturbation raised by the numerical scheme and operator approximation. Numerical experiments on the diffusion equation and Navier-Stokes equation show significant accuracy improvement compared with other unsupervised baselines, especially for the vibrated initial condition and long-time simulation settings.

Multilingual Speech Emotion Recognition With Multi-Gating Mechanism and Neural Architecture Search

Speech emotion recognition (SER) classifies audio into emotion categories such as Happy, Angry, Fear, Disgust and Neutral. While Speech Emotion Recognition (SER) is a common application for popular languages, it continues to be a problem for low-resourced languages, i.e., languages with no pretrained speech-to-text recognition models. This paper firstly proposes a language-specific model that extract emotional information from multiple pre-trained speech models, and then designs a multi-domain model that simultaneously performs SER for various languages. Our multidomain model employs a multi-gating mechanism to generate unique weighted feature combination for each language, and also searches for specific neural network structure for each language through a neural architecture search module. In addition, we introduce a contrastive auxiliary loss to build more separable representations for audio data. Our experiments show that our model raises the state-of-the-art accuracy by 3% for German and 14.3% for French.

Provable Adaptivity in Adam

Adaptive Moment Estimation (Adam) optimizer is widely used in deep learning tasks because of its fast convergence properties. However, the convergence of Adam is still not well understood. In particular, the existing analysis of Adam cannot clearly demonstrate the advantage of Adam over SGD. We attribute this theoretical embarrassment to $L$-smooth condition (i.e., assuming the gradient is globally Lipschitz continuous with constant $L$) adopted by literature, which has been pointed out to often fail in practical neural networks. To tackle this embarrassment, we analyze the convergence of Adam under a relaxed condition called $(L_0,L_1)$ smoothness condition, which allows the gradient Lipschitz constant to change with the local gradient norm. $(L_0,L_1)$ is strictly weaker than $L$-smooth condition and it has been empirically verified to hold for practical deep neural networks. Under the $(L_0,L_1)$ smoothness condition, we establish the convergence for Adam with practical hyperparameters. Specifically, we argue that Adam can adapt to the local smoothness condition, justifying the \emph{adaptivity} of Adam. In contrast, SGD can be arbitrarily slow under this condition. Our result might shed light on the benefit of adaptive gradient methods over non-adaptive ones.

Deep Random Vortex Method for Simulation and Inference of Navier-Stokes Equations

Navier-Stokes equations are significant partial differential equations that describe the motion of fluids such as liquids and air. Due to the importance of Navier-Stokes equations, the development on efficient numerical schemes is important for both science and engineer. Recently, with the development of AI techniques, several approaches have been designed to integrate deep neural networks in simulating and inferring the fluid dynamics governed by incompressible Navier-Stokes equations, which can accelerate the simulation or inferring process in a mesh-free and differentiable way. In this paper, we point out that the capability of existing deep Navier-Stokes informed methods is limited to handle non-smooth or fractional equations, which are two critical situations in reality. To this end, we propose the \emph{Deep Random Vortex Method} (DRVM), which combines the neural network with a random vortex dynamics system equivalent to the Navier-Stokes equation. Specifically, the random vortex dynamics motivates a Monte Carlo based loss function for training the neural network, which avoids the calculation of derivatives through auto-differentiation. Therefore, DRVM not only can efficiently solve Navier-Stokes equations involving rough path, non-differentiable initial conditions and fractional operators, but also inherits the mesh-free and differentiable benefits of the deep-learning-based solver. We conduct experiments on the Cauchy problem, parametric solver learning, and the inverse problem of both 2-d and 3-d incompressible Navier-Stokes equations. The proposed method achieves accurate results for simulation and inference of Navier-Stokes equations. Especially for the cases that include singular initial conditions, DRVM significantly outperforms existing PINN method.

Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEs

Stochastic partial differential equations (SPDEs) are significant tools for modeling dynamics in many areas including atmospheric sciences and physics. Neural Operators, generations of neural networks with capability of learning maps between infinite-dimensional spaces, are strong tools for solving parametric PDEs. However, they lack the ability to modeling SPDEs which usually have poor regularity due to the driving noise. As the theory of regularity structure has achieved great successes in analyzing SPDEs and provides the concept model feature vectors that well-approximate SPDEs' solutions, we propose the Neural Operator with Regularity Structure (NORS) which incorporates the feature vectors for modeling dynamics driven by SPDEs. We conduct experiments on various of SPDEs including the dynamic Phi41 model and the 2d stochastic Navier-Stokes equation, and the results demonstrate that the NORS is resolution-invariant, efficient, and achieves one order of magnitude lower error with a modest amount of data.