# Self-Composing Neural Operators with Depth and Accuracy Scaling via Adaptive Train-and-Unroll Approach

## TL;DR

This paper introduces a novel adaptive train-and-unroll framework for neural operators that enhances their depth, accuracy, and efficiency, achieving state-of-the-art results in scientific computing tasks.

## Contribution

It proposes a self-composing neural operator architecture with a depth scaling law and an adaptive training method, improving efficiency and accuracy without increasing model complexity.

## Key findings

- Achieves state-of-the-art performance on benchmark tasks.
- Demonstrates superior resolution of complex wave phenomena in USCT.
- Reveals an accuracy scaling law with model depth.

## Abstract

In this work, we propose a novel framework to enhance the efficiency and accuracy of neural operators through self-composition, offering both theoretical guarantees and practical benefits. Inspired by iterative methods in solving numerical partial differential equations (PDEs), we design a specific neural operator by repeatedly applying a single neural operator block, we progressively deepen the model without explicitly adding new blocks, improving the model's capacity. To train these models efficiently, we introduce an adaptive train-and-unroll approach, where the depth of the neural operator is gradually increased during training. This approach reveals an accuracy scaling law with model depth and offers significant computational savings through our adaptive training strategy. Our architecture achieves state-of-the-art (SOTA) performance on standard benchmarks. We further demonstrate its efficacy on a challenging high-frequency ultrasound computed tomography (USCT) problem, where a multigrid-inspired backbone enables superior performance in resolving complex wave phenomena. The proposed framework provides a computationally tractable, accurate, and scalable solution for large-scale data-driven scientific machine learning applications.