Do Neural Operators Forget Geometry? The Forgetting Hypothesis in Deep Operator Learning

• URL: http://arxiv.org/abs/2605.05862v1
• Date: Thu, 07 May 2026 08:31:33 GMT
• Title: Do Neural Operators Forget Geometry? The Forgetting Hypothesis in Deep Operator Learning
• Abstract summary: We show that neural operators perform well on structured domains, yet their behaviour on irregular geometries remains poorly understood.<n>We formalise the Geometric Forgetting Hypothesis: due to the Markovian structure of operator layers, neural operators progressively lose access to domain geometry as depth increases.<n>We introduce a lightweight geometry memory injection mechanism that restores geometric constraints at intermediate depths with minimal architectural overhead.
• Score: 0.2581652423269287
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Neural operators perform well on structured domains, yet their behaviour on irregular geometries remains poorly understood. We show that this limitation is not merely an encoding issue, but a depth-wise failure mode inherent to deep operator architectures. We formalise the Geometric Forgetting Hypothesis: due to the Markovian structure of operator layers and their reliance on global mixing mechanisms, neural operators progressively lose access to domain geometry as depth increases. Using layer-wise geometric probing, we demonstrate that both spectral and attention-based operators systematically lose geometric fidelity. We show that this geometric forgetting degrades accuracy, stability, and generalisation. To counteract it, we introduce a lightweight geometry memory injection mechanism that restores geometric constraints at intermediate depths with minimal architectural overhead. This simple intervention consistently mitigates forgetting and exposes a geometric shortcut instability in transformer-based operators, revealing that geometric retention is a structural requirement rather than a design choice.

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