Decoding Neural Networks Through Hamilton-Jacobi Equations

Training neural networks is routinely viewed through the lens of gradient descent. But visualize this: it's actually akin to solving Hamilton-Jacobi equations. Each gradient step is a choice, a selection of initial data for a viscous Hamilton-Jacobi equation. That's where the magic happens. The input during inference is the spatial point evaluated by this solution, and the initial condition is encoded in the weights.

The Structural Correspondence

This correspondence isn't just theoretical. It's spot on for log-sum-exp layers and structurally significant for broader architectures. Whether dealing with residual networks, transformers, or recurrent architectures, the same class of Hamilton-Jacobi equations is discretized, albeit with architecture-dependent Hamiltonian and viscosity. In simpler terms, the architecture you choose shapes the equation you're solving.

Unifying Perspectives

Here's where it gets intriguing. A single deformation parameter, ε, ties together four perspectives: network, tropical algebra, viscous PDE, and convex optimization. This commutative diagram unifies them under Lipschitz conditions. Numbers in context: the minimax optimal generalization rate hits at O(n^(-1/(d+2))) for a fixed t. Adversarial robustness? Controlled by ε. Backpropagation takes on a new role, acting as the co-state equation of the Hamiltonian system for residual networks.

Implications and Questions

What does all this mean? For one, it provides a new angle to evaluate neural network performance and robustness. Scaling exponents align with data intrinsic dimensions via PDE quadrature. A closed-form O(N) influence function emerges, where softmax attribution weights π_j reveal an entropy landscape undergoing fold bifurcations as ε increases. Why should we care about attribution basins merging? Because it impacts how we interpret model decisions.

The trend is clearer when you see it. Neural networks aren't just black boxes. They're mathematical structures with a deep connection to PDEs, offering insights into both robustness and interpretability. If ε holds so much sway over model behavior, shouldn't we pay closer attention to it? In the rush to build bigger and faster models, this perspective might just offer the clarity we need.