KO: Kinetics-inspired Neural Optimizer with PDE Simulation Approaches

TL;DR

KO introduces a physics-inspired optimizer that treats neural parameters as particles evolving under a Boltzmann Transport Equation framework, solved with Direct Simulation Monte Carlo. By applying hard and soft collision updates to gradients before the base optimizer step, KO promotes parameter dispersion and mitigates neuron condensation, with theoretical support via weight-cosine analysis and an H-theorem–style physical interpretation. Empirically, KO yields accuracy gains across image and text tasks (e.g., CIFAR-10/100, ImageNet, IMDB, Snips) while maintaining comparable compute to standard optimizers. This approach offers a principled, physics-grounded alternative to purely gradient-based methods and suggests avenues for efficient, hardware-friendly implementations in large-scale models.

Abstract

The design of optimization algorithms for neural networks remains a critical challenge, with most existing methods relying on heuristic adaptations of gradient-based approaches. This paper introduces KO (Kinetics-inspired Optimizer), a novel neural optimizer inspired by kinetic theory and partial differential equation (PDE) simulations. We reimagine the training dynamics of network parameters as the evolution of a particle system governed by kinetic principles, where parameter updates are simulated via a numerical scheme for the Boltzmann transport equation (BTE) that models stochastic particle collisions. This physics-driven approach inherently promotes parameter diversity during optimization, mitigating the phenomenon of parameter condensation, i.e. collapse of network parameters into low-dimensional subspaces, through mechanisms analogous to thermal diffusion in physical systems. We analyze this property, establishing both a mathematical proof and a physical interpretation. Extensive experiments on image classification (CIFAR-10/100, ImageNet) and text classification (IMDB, Snips) tasks demonstrate that KO consistently outperforms baseline optimizers (e.g., Adam, SGD), achieving accuracy improvements while computation cost remains comparable.

Figures (4)

• Figure 1: The architecture of KO. The upper part is the workflow of KO composed of a kinetic module and a base optimizer. The gradient is first calculated by network backpropagation, and then updated by the kinetic module, and finally fed into the base optimizer. The lower part is the kinetic module, which simulates the particle collision to update the gradient. The weight and gradient are viewed as the position and velocity of the particles with random collisions. Once two particles collide, the velocity of the particles is updated by the hard-body collision model.
• Figure 2: Condensation of two-layer NNs. The color indicates the cosine similarity of two hidden neurons’ input weights at epoch 100, whose indexes are indicated by the abscissa and the ordinate, respectively. The activation functions are indicated by the sub-captions. The first row shows the model weights trained with the original Adam optimizer. The second and third rows depict the results with the Soft Collision and Hard Collision, respectively.
• Figure 3: Accuracy and neuron similarity of three different models on CIFAR-100. The upper row shows the test accuracy while the lower row depicts the neuron similarity changes during training.
• Figure 4: Condensation of Resnet18-like neural networks on CIFAR-10. The color in the figures indicates the cosine similarity of the normalized input weights of two neurons in the first FC layer. The subcaption represents the activation function in the FC layers. The first column shows the model weights trained with the original Adam optimizer. The second and third columns depict the results with the Soft Collision and Hard Collision, respectively.

Theorems & Definitions (8)

• Theorem 3.1: dziugaite2017computingnonvacuousgeneralizationbounds
• Theorem 3.2: jin2020doesweightcorrelationaffect
• Theorem 3.3
• Definition 3.4: H quantity
• Definition 3.5: Entropy $S$
• Lemma 3.6
• Theorem 3.7: $H$-theorem
• proof : Proof of Thm. \ref{['thm:3']}