# TRADE: Transfer of Distributions between External Conditions with Normalizing Flows

arXiv: https://arxiv.org/abs/2410.19492 · HTML: https://mlsanity.com/paper/2410.19492/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 58 references that are in the map.
- First posted on arXiv 25 Oct 2024.

## Builds on (most cited first, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Automatic Posterior Transformation for Likelihood-Free Inference](https://mlsanity.com/paper/1905.07488/) (2019)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [i-RevNet: Deep Invertible Networks](https://mlsanity.com/paper/1802.07088/) (2018)
- [Disentanglement by Nonlinear ICA with General Incompressible-flow Networks (GIN)](https://mlsanity.com/paper/2001.04872/) (2020)
- [Targeted free energy estimation via learned mappings](https://mlsanity.com/paper/2002.04913/) (2020)
- [Augmented Normalizing Flows: Bridging the Gap Between Generative Flows and Latent Variable Models](https://mlsanity.com/paper/2002.07101/) (2020)
- [Advances in Black-Box VI: Normalizing Flows, Importance Weighting, and Optimization](https://mlsanity.com/paper/2006.10343/) (2020)
- [Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow models](https://mlsanity.com/paper/2108.12956/) (2021)
- [Free-form Flows: Make Any Architecture a Normalizing Flow](https://mlsanity.com/paper/2310.16624/) (2023)
- [Lifting Architectural Constraints of Injective Flows](https://mlsanity.com/paper/2306.01843/) (2023)
- [BR-SNIS: Bias Reduced Self-Normalized Importance Sampling](https://mlsanity.com/paper/2207.06364/) (2022)
- [Designing losses for data-free training of normalizing flows on Boltzmann distributions](https://mlsanity.com/paper/2301.05475/) (2023)
- [Physics-Informed Real NVP for Satellite Power System Fault Detection](https://mlsanity.com/paper/2405.17339/) (2024)
- [PINF: Continuous Normalizing Flows for Physics-Constrained Deep Learning](https://mlsanity.com/paper/2309.15139/) (2023)

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

- [Temperature-Annealed Boltzmann Generators](https://mlsanity.com/paper/2501.19077/) (2025)
- [Learning Boltzmann Generators via Constrained Mass Transport](https://mlsanity.com/paper/2510.18460/) (2025)
- [Efficient Training of Boltzmann Generators Using Off-Policy Log-Dispersion Regularization](https://mlsanity.com/paper/2602.03729/) (2026)

## Abstract (arXiv)

Modeling distributions that depend on external control parameters is a common scenario in diverse applications like molecular simulations, where system properties like temperature affect molecular configurations. Despite the relevance of these applications, existing solutions are unsatisfactory as they require severely restricted model architectures or rely on energy-based training, which is prone to instability. We introduce TRADE, which overcomes these limitations by formulating the learning process as a boundary value problem. By initially training the model for a specific condition using either i.i.d.~samples or backward KL training, we establish a boundary distribution. We then propagate this information across other conditions using the gradient of the unnormalized density with respect to the external parameter. This formulation, akin to the principles of physics-informed neural networks, allows us to efficiently learn parameter-dependent distributions without restrictive assumptions. Experimentally, we demonstrate that TRADE achieves excellent results in a wide range of applications, ranging from Bayesian inference and molecular simulations to physical lattice models.