# F-Adapter: Frequency-Adaptive Parameter-Efficient Fine-Tuning in Scientific Machine Learning

arXiv: https://arxiv.org/abs/2509.23173 · HTML: https://mlsanity.com/paper/2509.23173/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 62 references that are in the map.
- First posted on arXiv 27 Sep 2025.

## Builds on (most cited first, 30 of 48)

- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [Prefix-Tuning: Optimizing Continuous Prompts for Generation](https://mlsanity.com/paper/2101.00190/) (2021)
- [The Power of Scale for Parameter-Efficient Prompt Tuning](https://mlsanity.com/paper/2104.08691/) (2021)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [BitFit: Simple Parameter-efficient Fine-tuning for Transformer-based Masked Language-models](https://mlsanity.com/paper/2106.10199/) (2021)
- [Few-Shot Parameter-Efficient Fine-Tuning is Better and Cheaper than In-Context Learning](https://mlsanity.com/paper/2205.05638/) (2022)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [AdaptFormer: Adapting Vision Transformers for Scalable Visual Recognition](https://mlsanity.com/paper/2205.13535/) (2022)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [AdaLoRA: Adaptive Budget Allocation for Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2303.10512/) (2023)
- [Parameter-Efficient Transfer Learning with Diff Pruning](https://mlsanity.com/paper/2012.07463/) (2020)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)
- [Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior](https://mlsanity.com/paper/2306.00258/) (2023)
- [Scalable Transformer for PDE Surrogate Modeling](https://mlsanity.com/paper/2305.17560/) (2023)
- [HydraLoRA: An Asymmetric LoRA Architecture for Efficient Fine-Tuning](https://mlsanity.com/paper/2404.19245/) (2024)
- [Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximation](https://mlsanity.com/paper/2405.07200/) (2024)
- [Multiple Physics Pretraining for Physical Surrogate Models](https://mlsanity.com/paper/2310.02994/) (2023)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.11833/) (2023)
- [Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs](https://mlsanity.com/paper/2403.12553/) (2024)
- [SVFT: Parameter-Efficient Fine-Tuning with Singular Vectors](https://mlsanity.com/paper/2405.19597/) (2024)
- [Polyhistor: Parameter-Efficient Multi-Task Adaptation for Dense Vision Tasks](https://mlsanity.com/paper/2210.03265/) (2022)
- [Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning](https://mlsanity.com/paper/2206.10745/) (2022)

## Built on by (most cited first, 3 of 3)

- [Iterative Refinement Neural Operators are Learned Fixed-Point Solvers: A Principled Approach to Spectral Bias Mitigation](https://mlsanity.com/paper/2605.24041/) (2026)
- [Adapting Automotive Aerodynamics Surrogates to New Vehicle Families via Transfer Learning](https://mlsanity.com/paper/2605.27968/) (2026)
- [A Model-Agnostic Physics-Guided Adapter for Few-Shot Transfer of Coastal Flood Prediction Models to Unseen Regions](https://mlsanity.com/paper/2609.37565/) (2026)

## Abstract (arXiv)

Parameter-efficient fine-tuning (PEFT) of powerful pre-trained models for complex downstream tasks has proven effective in vision and language processing, yet this paradigm remains unexplored in scientific machine learning, where the objective is to model complex physical systems. We conduct the first systematic study of PEFT for pre-trained Large Operator Models (LOMs) obtained by scaling variants of Fourier Neural Operator. First, we observe that the widely used Low-Rank Adaptation (LoRA) yields markedly poorer performance on LOMs than Adapter tuning. Then, we further theoretically establish that stacked LoRA incurs a depth-amplified lower bound on approximation error within Fourier layers, whereas adapters retain universal approximation capacity and, by concentrating parameters on energy-dominant low-frequency modes, attain exponentially decaying error with bottleneck width in the Fourier domain. Motivated by the robust empirical gains of adapters and by our theoretical characterization of PDE solutions as spectrally sparse, we introduce Frequency-Adaptive Adapter (F-Adapter). F-Adapter allocates adapter capacity based on spectral complexity, assigning higher-dimension modules to low-frequency components and lower-dimension modules to high-frequency components. Our F-Adapters establish state-of-the-art (SOTA) results on multiple challenging 3D Navier-Stokes benchmarks, markedly enhancing both generalization and spectral fidelity over LoRA and other PEFT techniques commonly used in LLMs. To the best of our knowledge, this work is the first to explore PEFT for scientific machine-learning and establishes F-Adapter as an effective paradigm for this domain.