# Differential Machine Learning

arXiv: https://arxiv.org/abs/2005.02347 · HTML: https://mlsanity.com/paper/2005.02347/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 May 2020.

## Built on by (most cited first, 10 of 10)

- [A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations](https://mlsanity.com/paper/2404.08456/) (2024)
- [Applying Deep Learning to Calibrate Stochastic Volatility Models](https://mlsanity.com/paper/2309.07843/) (2023)
- [Parameterized Neural Networks for Finance](https://mlsanity.com/paper/2304.08883/) (2023)
- [Towards Sobolev Pruning](https://mlsanity.com/paper/2312.03510/) (2023)
- [Harnessing the Power of Gradient-Based Simulations for Multi-Objective Optimization in Particle Accelerators](https://mlsanity.com/paper/2411.04817/) (2024)
- [Differential Machine Learning for Time Series Prediction](https://mlsanity.com/paper/2503.03302/) (2025)
- [Hybrid least squares for learning functions from highly noisy data](https://mlsanity.com/paper/2507.02215/) (2025)
- [Parametric Numerical Integration with (Differential) Machine Learning](https://mlsanity.com/paper/2512.11530/) (2025)
- [Amortizing the Calibration Triple: A Projection-Consistent Neural Operator for Local-Stochastic Volatility](https://mlsanity.com/paper/2608.01217/) (2026)
- [Beyond Time Induction: Structured Conditional Neural Solvers for Parametric Stochastic-Volatility Pricing](https://mlsanity.com/paper/2610.19254/) (2026)

## Abstract (arXiv)

Differential machine learning combines automatic adjoint differentiation (AAD) with modern machine learning (ML) in the context of risk management of financial Derivatives. We introduce novel algorithms for training fast, accurate pricing and risk approximations, online, in real-time, with convergence guarantees. Our machinery is applicable to arbitrary Derivatives instruments or trading books, under arbitrary stochastic models of the underlying market variables. It effectively resolves computational bottlenecks of Derivatives risk reports and capital calculations. Differential ML is a general extension of supervised learning, where ML models are trained on examples of not only inputs and labels but also differentials of labels wrt inputs. It is also applicable in many situations outside finance, where high quality first-order derivatives wrt training inputs are available. Applications in Physics, for example, may leverage differentials known from first principles to learn function approximations more effectively. In finance, AAD computes pathwise differentials with remarkable efficacy so differential ML algorithms provide extremely effective pricing and risk approximations. We can produce fast analytics in models too complex for closed form solutions, extract the risk factors of complex transactions and trading books, and effectively compute risk management metrics like reports across a large number of scenarios, backtesting and simulation of hedge strategies, or regulations like XVA, CCR, FRTB or SIMM-MVA. TensorFlow implementation is available on https://github.com/differential-machine-learning