Ábel Ságodi

New preprint from the lab! Ábel Ságodi developed a theory of approximating dynamical systems that goes beyond finite time. #theoreticalNeuroscience
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Universal Approximation Theorems for Dynamical Systems with Infinite-Time Horizon Guarantees. . arxiv.org/abs/2602.08640

(TL;DR) If you are modeling neural computation over long time scales, "Fading Memory" is a fundamental limitation.

We provide the theoretical framework to ensure your model can capture the memories, decisions, and rhythms that actually matter.

#NeuroAI #DynamicalSystems #NeuralODEs (6/6)

Subsequently, we can guarantee a temporal generalization error bound for a given precision and reliability.

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Result: We prove you can approximate these dynamics with arbitrary precision and arbitrary reliability.

1️⃣ ε (Precision): Maximum allowed trajectory tracking error
2️⃣ δ (Reliability): How small to shrink the B-type error

We prove a Neural ODE exists that satisfies both constraints forever.

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Why not just "train longer"? You hit topological walls.

We identify three specific failure modes for infinite-time dynamics:

1️⃣ B-type: Tiny errors near a decision boundary switch the outcome.
2️⃣ P-type: Oscillations drift out of phase.
3️⃣ D-type: Continuous attractors break into points.

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Decision making & working memory require multistability—distinct basins of attraction to hold a choice or a continuous value.

If your model has Fading Memory (like liquid state machines), it must eventually drift back to a global baseline. It literally cannot hold a memory forever.

(2/6)

Can we guarantee the behavior of an RNN to generalize well for infinite time? 🧠♾️
Similar to universal approximation theorems in deep nets, for systems that forget everything eventually, there are guarantees. We prove it for multistable systems!
arxiv.org/abs/2602.08640 w/ @memming.bsky.social
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