# Fourier Amplitude and Correlation Loss: Beyond Using L2 Loss for Skillful Precipitation Nowcasting

arXiv: https://arxiv.org/abs/2410.23159 · HTML: https://mlsanity.com/paper/2410.23159/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 50 references that are in the map.
- First posted on arXiv 30 Oct 2024.

## Builds on (most cited first, 30 of 34)

- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [MAGVIT: Masked Generative Video Transformer](https://mlsanity.com/paper/2212.05199/) (2022)
- [Skillful Precipitation Nowcasting using Deep Generative Models of Radar](https://mlsanity.com/paper/2104.00954/) (2021)
- [MCVD: Masked Conditional Video Diffusion for Prediction, Generation, and Interpolation](https://mlsanity.com/paper/2205.09853/) (2022)
- [StyleGAN-V: A Continuous Video Generator with the Price, Image Quality and Perks of StyleGAN2](https://mlsanity.com/paper/2112.14683/) (2021)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [Stochastic Video Generation with a Learned Prior](https://mlsanity.com/paper/1802.07687/) (2018)
- [Focal Frequency Loss for Image Reconstruction and Synthesis](https://mlsanity.com/paper/2012.12821/) (2020)
- [Learning in the Frequency Domain](https://mlsanity.com/paper/2002.12416/) (2020)
- [Disentangling Physical Dynamics from Unknown Factors for Unsupervised Video Prediction](https://mlsanity.com/paper/2003.01460/) (2020)
- [SimVP: Simpler yet Better Video Prediction](https://mlsanity.com/paper/2206.05099/) (2022)
- [Deep Learning for Precipitation Nowcasting: A Benchmark and A New Model](https://mlsanity.com/paper/1706.03458/) (2017)
- [PredRNN: A Recurrent Neural Network for Spatiotemporal Predictive Learning](https://mlsanity.com/paper/2103.09504/) (2021)
- [Earthformer: Exploring Space-Time Transformers for Earth System Forecasting](https://mlsanity.com/paper/2207.05833/) (2022)
- [Diffusion Models for Video Prediction and Infilling](https://mlsanity.com/paper/2206.07696/) (2022)
- [PredRNN++: Towards A Resolution of the Deep-in-Time Dilemma in Spatiotemporal Predictive Learning](https://mlsanity.com/paper/1804.06300/) (2018)
- [PreDiff: Precipitation Nowcasting with Latent Diffusion Models](https://mlsanity.com/paper/2307.10422/) (2023)
- [Memory In Memory: A Predictive Neural Network for Learning Higher-Order Non-Stationarity from Spatiotemporal Dynamics](https://mlsanity.com/paper/1811.07490/) (2018)
- [Temporal Attention Unit: Towards Efficient Spatiotemporal Predictive Learning](https://mlsanity.com/paper/2206.12126/) (2022)
- [Fourier Space Losses for Efficient Perceptual Image Super-Resolution](https://mlsanity.com/paper/2106.00783/) (2021)
- [SmaAt-UNet: Precipitation Nowcasting using a Small Attention-UNet Architecture](https://mlsanity.com/paper/2007.04417/) (2020)
- [DiffCast: A Unified Framework via Residual Diffusion for Precipitation Nowcasting](https://mlsanity.com/paper/2312.06734/) (2023)
- [SAPE: Spatially-Adaptive Progressive Encoding for Neural Optimization](https://mlsanity.com/paper/2104.09125/) (2021)
- [Latent diffusion models for generative precipitation nowcasting with accurate uncertainty quantification](https://mlsanity.com/paper/2304.12891/) (2023)
- [OpenSTL: A Comprehensive Benchmark of Spatio-Temporal Predictive Learning](https://mlsanity.com/paper/2306.11249/) (2023)
- [STRPM: A Spatiotemporal Residual Predictive Model for High-Resolution Video Prediction](https://mlsanity.com/paper/2203.16084/) (2022)

## Built on by (most cited first, 16 of 16)

- [WaveC2R: Wavelet-Driven Coarse-to-Refined Hierarchical Learning for Radar Retrieval](https://mlsanity.com/paper/2511.17558/) (2025)
- [Extreme Weather Nowcasting via Local Precipitation Pattern Prediction](https://mlsanity.com/paper/2602.05204/) (2026)
- [DuoCast: Duo-Probabilistic Diffusion for Precipitation Nowcasting](https://mlsanity.com/paper/2412.01091/) (2024)
- [Observation-guided Interpolation Using Graph Neural Networks for High-Resolution Nowcasting in Switzerland](https://mlsanity.com/paper/2509.00017/) (2025)
- [Assessing the Geographic Generalization and Physical Consistency of Generative Models for Climate Downscaling](https://mlsanity.com/paper/2510.13722/) (2025)
- [RainPro-8: An Efficient Deep Learning Model to Estimate Rainfall Probabilities Over 8 Hours](https://mlsanity.com/paper/2505.10271/) (2025)
- [LangPrecip: Language-Aware Multimodal Precipitation Nowcasting](https://mlsanity.com/paper/2512.22317/) (2025)
- [McCast: Memory-Guided Latent Drift Correction for Long-Horizon Precipitation Nowcasting](https://mlsanity.com/paper/2605.13197/) (2026)
- [RainODE: Continuous-Time Precipitation Forecasting with Latent Neural ODEs](https://mlsanity.com/paper/2606.29855/) (2026)
- [STLDM: Spatio-Temporal Latent Diffusion Model for Precipitation Nowcasting](https://mlsanity.com/paper/2512.21118/) (2025)
- [Field-Space Autoencoder for Scalable Climate Emulators](https://mlsanity.com/paper/2601.15102/) (2026)
- [WADEPre: A Wavelet-based Decomposition Model for Extreme Precipitation Nowcasting with Multi-Scale Learning](https://mlsanity.com/paper/2602.02096/) (2026)
- [MambaRain: Multi-Scale Mamba-Attention Framework for 0-3 Hour Precipitation Nowcasting](https://mlsanity.com/paper/2605.14606/) (2026)
- [FreCast: Refining Radar Echo Intensity via Phase-Preserving Amplitude Residual Diffusion for Precipitation Nowcasting](https://mlsanity.com/paper/2608.08436/) (2026)
- [Diffusion-Based Refinement for Kilometer-Scale Probabilistic Precipitation Nowcasting](https://mlsanity.com/paper/2608.30205/) (2026)
- [A Free Knob: Decoupling Calibration and Predictive Skill in Threshold-Based Evaluation](https://mlsanity.com/paper/2609.33457/) (2026)

## Abstract (arXiv)

Deep learning approaches have been widely adopted for precipitation nowcasting in recent years. Previous studies mainly focus on proposing new model architectures to improve pixel-wise metrics. However, they frequently result in blurry predictions which provide limited utility to forecasting operations. In this work, we propose a new Fourier Amplitude and Correlation Loss (FACL) which consists of two novel loss terms: Fourier Amplitude Loss (FAL) and Fourier Correlation Loss (FCL). FAL regularizes the Fourier amplitude of the model prediction and FCL complements the missing phase information. The two loss terms work together to replace the traditional L2 losses such as MSE and weighted MSE for the spatiotemporal prediction problem on signal-based data. Our method is generic, parameter-free and efficient. Extensive experiments using one synthetic dataset and three radar echo datasets demonstrate that our method improves perceptual metrics and meteorology skill scores, with a small trade-off to pixel-wise accuracy and structural similarity. Moreover, to improve the error margin in meteorological skill scores such as Critical Success Index (CSI) and Fractions Skill Score (FSS), we propose and adopt the Regional Histogram Divergence (RHD), a distance metric that considers the patch-wise similarity between signal-based imagery patterns with tolerance to local transforms. Code is available at https://github.com/argenycw/FACL