A.1 Difference between the performance of two joint policies In Section 3.1, the difference between the performance of two joint policies is expressed as follows: The proof is a multi-agent version of the proof in (Kakade and Langford, 2002). Now we provide the mathematical detail formally. A.2 Approximation that matches the true value to first order In Section 3.1, we claim that Jπ( π) matches J( π) to first order. Intuitively, this means that a sufficiently small update of the joint policy which improves Jπ( π) will also improve J( π). Now we prove it formally.

Understanding the learning process of artificial neural networks requires clarifying the structure of the parameter space within which learning takes place. A neural network parameter's functional equivalence class is the set of parameters implementing the same input-output function. For many architectures, almost all parameters have a simple and well-documented functional equivalence class. However, there is also a vanishing minority of reducible parameters, with richer functional equivalence classes caused by redundancies among the network's units. In this paper, we give an algorithmic characterisation of unit redundancies and reducible functional equivalence classes for a single-hidden-layer hyperbolic tangent architecture. We show that such functional equivalence classes are piecewise-linear path-connected sets, and that for parameters with a majority of redundant units, the sets have a diameter of at most 7 linear segments.

We introduce a probability distribution, combined with an efficient sampling algorithm, for weights and biases of fully-connected neural networks.

Neural Information Processing Systems http://nips.cc/

In this article, we continue this line of investigation.

In the last years, neural networks (NN) have evolved from laboratory environments to the state-of-the-art for many real-world problems. It was shown that NN models (i.e., their weights and biases) evolve on unique trajectories in weight space during training. Following, a population of such neural network models (referred to as model zoo) would form structures in weight space. We think that the geometry, curvature and smoothness of these structures contain information about the state of training and can reveal latent properties of individual models. With such model zoos, one could investigate novel approaches for (i) model analysis, (ii) discover unknown learning dynamics, (iii) learn rich representations of such populations, or (iv) exploit the model zoos for generative modelling of NN weights and biases.

In physics, especially in geosciences and climate sciences, the poor performance of neural networks (NN) when applied outside their training distribution or their trained dynamics poses a very strong limitation to their general applicability (Irrgang et al., 2021; Landsberg and Barnes, 2025). In these fields, physical relations such as laws, dependencies or sensitivities are commonly derived (or learned) under well observed conditions and are then applied to less observed conditions to gain knowledge about the latter. For example, results from lab or numerical model experiments are regularly applied to real world problems or observations (e.g., Mehta et al., 2025); knowledge from our Earth and our Solar System are transferred to other planets and other star systems (e.g., Kvorka et al., 2026); learned relations that are derived today are transferred to the distant past or to the future (e.g., Eyring et al., 2016; Wang et al., 2024; Koutsodendris et al., 2014).

Neural Information Processing Systems http://nips.cc/