Christopher P. Monterola

Jose Marie Antonio Miñoza, Erika Fille T. Legara, Christopher P. Monterola

In this paper, training a neural network is identified, exactly, as a search through Hamilton--Jacobi initial-value problems: each gradient step selects the initial data of a viscous Hamilton--Jacobi equation whose Hopf--Cole propagator best fits the observations; at inference, the input is the spatial point at which that solution is evaluated and the initial condition is already encoded in the weights. The correspondence is exact for log-sum-exp layers and structural for broader architectures: residual networks, transformers, and recurrent architectures (RNNs, LSTMs, SSMs) each discretize the same class of Hamilton--Jacobi equations, with architecture-dependent Hamiltonian and viscosity. A single deformation parameter $\varepsilon$ unifies all four perspectives (network, tropical algebra, viscous PDE, convex optimization) in a commutative diagram closed under Lipschitz conditions. Quantitative consequences include: the minimax optimal generalization rate $O(n^{-1/(d+2)})$ for fixed $t$; adversarial robustness controlled by $\varepsilon$; backpropagation as the co-state equation of the Hamiltonian system for residual networks (Pontryagin Maximum Principle); scaling exponents consistent with data intrinsic dimension via PDE quadrature; and a closed-form $O(N)$ influence function (softmax attribution weights $π_j$) whose entropy landscape undergoes fold bifurcations as $\varepsilon$ increases, each merging attribution basins.

Kristine Ann M. Carandang, Jasper Meynard P. Araña, Ethan Robert A. Casin et al.

Due to the legal and ethical responsibilities of healthcare providers (HCPs) for accurate documentation and protection of patient data privacy, the natural variability in the responses of large language models (LLMs) presents challenges for incorporating clinical note generation (CNG) systems, driven by LLMs, into real-world clinical processes. The complexity is further amplified by the detailed nature of texts in CNG. To enhance the confidence of HCPs in tools powered by LLMs, this study evaluates the reliability of 12 open-weight and proprietary LLMs from Anthropic, Meta, Mistral, and OpenAI in CNG in terms of their ability to generate notes that are string equivalent (consistency rate), have the same meaning (semantic consistency) and are correct (semantic similarity), across several iterations using the same prompt. The results show that (1) LLMs from all model families are stable, such that their responses are semantically consistent despite being written in various ways, and (2) most of the LLMs generated notes close to the corresponding notes made by experts. Overall, Meta's Llama 70B was the most reliable, followed by Mistral's Small model. With these findings, we recommend the local deployment of these relatively smaller open-weight models for CNG to ensure compliance with data privacy regulations, as well as to improve the efficiency of HCPs in clinical documentation.