Recently, the anchor acceleration, an acceleration mechanism distinct from Nes-terov's, has been discovered for minimax optimization and fixed-point problems,

Can we apply this analysis to transformers?

Understanding the geometric properties of gradient descent dynamics is a key ingredient in deciphering the recent success of very large machine learning models. A striking observation is that trained over-parameterized models retain some properties of the optimization initialization. This implicit bias is believed to be responsible for some favorable properties of the trained models and could explain their good generalization properties. The purpose of this article is threefold. First, we rigorously expose the definition and basic properties of conservation laws, that define quantities conserved during gradient flows of a given model (e.g. of a ReLU network with a given architecture) with any training data and any loss.

Neural operators have demonstrated considerable effectiveness in accelerating the solution of time-dependent partial differential equations (PDEs) by directly learning governing physical laws from data. However, for PDEs governed by conservation laws(e.g., conservation of mass, energy, or matter), existing neural operators fail to satisfy conservation properties, which leads to degraded model performance and limited generalizability. Moreover, we observe that distinct PDE problems generally require different optimal neural network architectures. This finding underscores the inherent limitations of specialized models in generalizing across diverse problem domains. To address these limitations, we propose Exterior-Embedded Conservation Framework (ECF), a universal conserving framework that can be integrated with various data-driven neural operators to enforce conservation laws strictly in predictions. The framework consists of two key components: a conservation quantity encoder that extracts conserved quantities from input data, and a conservation quantity decoder that adjusts the neural operator's predictions using these quantities to ensure strict conservation compliance in the final output. Since our architecture enforces conservation laws, we theoretically prove that it enhances model performance. To validate the performance of our method, we conduct experiments on multiple conservation-law-constrained PDE scenarios, including adiabatic systems, shallow water equations, and the Allen-Cahn problem. These baselines demonstrate that our method effectively improves model accuracy while strictly enforcing conservation laws in the predictions.

Standard physics-informed neural network implementations have produced large error rates when using these models to solve the regularized long wave (RLW) equation. Two improved PINN approaches were developed in this research: an adaptive approach with self-adaptive loss weighting and a conservative approach enforcing explicit conservation laws. Three benchmark tests were used to demonstrate how effective PINN's are as they relate to the type of problem being solved (i.e., time dependent RLW equation). The first was a single soliton traveling along a line (propagation), the second was the interaction between two solitons, and the third was the evolution of an undular bore over the course of $t=250$. The results demonstrated that the effectiveness of PINNs are problem specific. The adaptive PINN was significantly better than both the conservative PINN and the standard PINN at solving problems involving complex nonlinear interactions such as colliding two solitons. The conservative approach was significantly better at solving problems involving long term behavior of single solitons and undular bores. However, the most important finding from this research is that explicitly enforcing conservation laws may be harmful to optimizing the solution of highly nonlinear systems of equations and therefore requires special training methods. The results from our adaptive and conservative approaches were within $O(10^{-5})$ of established numerical solutions for the same problem, thus demonstrating that PINNs can provide accurate solutions to complex systems of partial differential equations without the need for a discretization of space or time (mesh free). Moreover, the finding from this research challenges the assumptions that conservation enforcement will always improve the performance of a PINN and provides researchers with guidelines for designing PINNs for use on specific types of problems.

How might we endow them with better inductive biases?

In this work, we focus on sequential prediction problems and take inspiration from Noether's theorem to reduce the