# Unbiased deep solvers for linear parametric PDEs

arXiv: https://arxiv.org/abs/1810.05094 · HTML: https://mlsanity.com/paper/1810.05094/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Oct 2018.

## Built on by (most cited first, 5 of 5)

- [Trust Region Constrained Measure Transport in Path Space for Stochastic Optimal Control and Inference](https://mlsanity.com/paper/2508.12511/) (2025)
- [FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing Flows and the Feynman-Kac Formula](https://mlsanity.com/paper/2503.11427/) (2025)
- [Prediction-Enhanced Monte Carlo: A Machine Learning View on Control Variate](https://mlsanity.com/paper/2412.11257/) (2024)
- [Practical Aspects on Solving Differential Equations Using Deep Learning: A Primer](https://mlsanity.com/paper/2408.11266/) (2024)
- [Deep Operator BSDE: a Numerical Scheme to Approximate Solution Operators](https://mlsanity.com/paper/2412.03405/) (2024)

## Abstract (arXiv)

We develop several deep learning algorithms for approximating families of parametric PDE solutions. The proposed algorithms approximate solutions together with their gradients, which in the context of mathematical finance means that the derivative prices and hedging strategies are computed simulatenously. Having approximated the gradient of the solution one can combine it with a Monte-Carlo simulation to remove the bias in the deep network approximation of the PDE solution (derivative price). This is achieved by leveraging the Martingale Representation Theorem and combining the Monte Carlo simulation with the neural network. The resulting algorithm is robust with respect to quality of the neural network approximation and consequently can be used as a black-box in case only limited a priori information about the underlying problem is available. We believe this is important as neural network based algorithms often require fair amount of tuning to produce satisfactory results. The methods are empirically shown to work for high-dimensional problems (e.g. 100 dimensions). We provide diagnostics that shed light on appropriate network architectures.