Countercurrent Multiplier Networks: A Renal-Inspired Iterative Operator with Provably Bounded Fixed-Point Dynamics

Snigdha Chandan Khilar

cs.LGJul 21, 2026

Countercurrent Multiplier Networks: A Renal-Inspired Iterative Operator with Provably Bounded Fixed-Point Dynamics

Authors: Snigdha Chandan Khilar

Organizations: Independent Researcher

Abstract

The mammalian kidney concentrates urine using a mechanism with no analogue in current neural architectures: the countercurrent multiplier. Two anti-parallel flows joined at a hairpin recirculate a weak magnitude-bounded local pump into a large axial gradient achieving a four-fold concentration increase from a single-effect gradient that never exceeds 200 mOsm at any point. We formalize this mechanism as a differentiable sequence operator the Countercurrent Multiplier (CCM) layer and study it as an alternative to residual iterative refinement.

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May 21, 2026 · cs.LGJ/K move · Enter open · S save

Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximations

Shuang Chen, Juncai He, Xue-Cheng Tai

Qiuzhen College, Tsinghua University, Haidian District, Beijing 100084, China · Yau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing 100084, China · Norwegian Research Center, Fantoftvegen 38, 5072 Bergen, Norway

We introduce an abstract neural flow framework for neural networks and neural operators. The framework contains two continuous-depth models, namely neural flows with composition and separation structures, and covers both finite-dimensional function approximation and infinite-dimensional operator approximation. We prove well-posedness and universal approximation properties for the corresponding neural flows, including, to the best of our knowledge, the first universal approximation result for flow-based models between infinite-dimensional spaces. We also obtain universal approximation results for convolutional neural flow models. Through suitable time discretizations, the composition structure recovers ResNet-type architectures, while the separation structure, via a splitting-based discretization, yields plain architectures. This gives a unified flow-based route to both residual and plain architectures for neural networks and neural operators with fully connected or convolutional linear layers.

Neural Operators Neural-Network Approximation Convolutional Neural Networks Approximation Abstraction

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Abstract

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