# Short-Term Density Forecasting of Low-Voltage Load using Bernstein-Polynomial Normalizing Flows

arXiv: https://arxiv.org/abs/2204.13939 · HTML: https://mlsanity.com/paper/2204.13939/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 69 references that are in the map.
- First posted on arXiv 29 Apr 2022.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Parallel WaveNet: Fast High-Fidelity Speech Synthesis](https://mlsanity.com/paper/1711.10433/) (2017)
- [Multivariate Probabilistic Time Series Forecasting via Conditioned Normalizing Flows](https://mlsanity.com/paper/2002.06103/) (2020)
- [Sum-of-Squares Polynomial Flow](https://mlsanity.com/paper/1905.02325/) (2019)
- [Conditional Density Estimation with Bayesian Normalising Flows](https://mlsanity.com/paper/1802.04908/) (2018)
- [Review of Smart Meter Data Analytics: Applications, Methodologies, and Challenges](https://mlsanity.com/paper/1802.04117/) (2018)
- [Cubic-Spline Flows](https://mlsanity.com/paper/1906.02145/) (2019)
- [Noise Regularization for Conditional Density Estimation](https://mlsanity.com/paper/1907.08982/) (2019)
- [Short-term Load Forecasting with Deep Residual Networks](https://mlsanity.com/paper/1805.11956/) (2018)
- [Most Likely Transformations](https://mlsanity.com/paper/1508.06749/) (2015)
- [Deep transformation models: Tackling complex regression problems with neural network based transformation models](https://mlsanity.com/paper/2004.00464/) (2020)
- [Review of Low Voltage Load Forecasting: Methods, Applications, and Recommendations](https://mlsanity.com/paper/2106.00006/) (2021)
- [Scenario Forecasting of Residential Load Profiles](https://mlsanity.com/paper/1906.07373/) (2019)
- [Robust normalizing flows using Bernstein-type polynomials](https://mlsanity.com/paper/2102.03509/) (2021)

## Built on by (most cited first, 6 of 6)

- [Sample-based Uncertainty Quantification with a Single Deterministic Neural Network](https://mlsanity.com/paper/2209.08418/) (2022)
- [QUPID: A Partitioned Quantum Neural Network for Anomaly Detection in Smart Grid](https://mlsanity.com/paper/2601.11500/) (2026)
- [Creating Probabilistic Forecasts from Arbitrary Deterministic Forecasts using Conditional Invertible Neural Networks](https://mlsanity.com/paper/2302.01800/) (2023)
- [AutoPQ: Automating Quantile estimation from Point forecasts in the context of sustainability](https://mlsanity.com/paper/2412.00419/) (2024)
- [Learning and Generating Diverse Residential Load Patterns Using GAN with Weakly-Supervised Training and Weight Selection](https://mlsanity.com/paper/2504.14300/) (2025)
- [Hybrid Bernstein Normalizing Flows for Flexible Multivariate Density Regression with Interpretable Marginals](https://mlsanity.com/paper/2505.14164/) (2025)

## Abstract (arXiv)

The transition to a fully renewable energy grid requires better forecasting of demand at the low-voltage level to increase efficiency and ensure reliable control. However, high fluctuations and increasing electrification cause huge forecast variability, not reflected in traditional point estimates. Probabilistic load forecasts take future uncertainties into account and thus allow more informed decision-making for the planning and operation of low-carbon energy systems. We propose an approach for flexible conditional density forecasting of short-term load based on Bernstein polynomial normalizing flows, where a neural network controls the parameters of the flow. In an empirical study with 363 smart meter customers, our density predictions compare favorably against Gaussian and Gaussian mixture densities. Also, they outperform a non-parametric approach based on the pinball loss for 24h-ahead load forecasting for two different neural network architectures.