# Neural Weighted A*: Learning Graph Costs and Heuristics with Differentiable Anytime A*

arXiv: https://arxiv.org/abs/2105.01480 · HTML: https://mlsanity.com/paper/2105.01480/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 36 references that are in the map.
- First posted on arXiv 4 May 2021.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Combinatorial Optimization with Reinforcement Learning](https://mlsanity.com/paper/1611.09940/) (2016)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Deep Learning on Graphs: A Survey](https://mlsanity.com/paper/1812.04202/) (2018)
- [Reinforcement Learning for Solving the Vehicle Routing Problem](https://mlsanity.com/paper/1802.04240/) (2018)
- [A Survey of Motion Planning and Control Techniques for Self-driving Urban Vehicles](https://mlsanity.com/paper/1604.07446/) (2016)
- [Differentiation of Blackbox Combinatorial Solvers](https://mlsanity.com/paper/1912.02175/) (2019)
- [Conditional Random Fields as Recurrent Neural Networks](https://mlsanity.com/paper/1502.03240/) (2015)
- [Differentiable MPC for End-to-end Planning and Control](https://mlsanity.com/paper/1810.13400/) (2018)
- [SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver](https://mlsanity.com/paper/1905.12149/) (2019)
- [Differentiable Rendering: A Survey](https://mlsanity.com/paper/2006.12057/) (2020)
- [Learning with Differentiable Perturbed Optimizers](https://mlsanity.com/paper/2002.08676/) (2020)
- [Path Planning using Neural A* Search](https://mlsanity.com/paper/2009.07476/) (2020)
- [Semantic Image Segmentation via Deep Parsing Network](https://mlsanity.com/paper/1509.02634/) (2015)
- [Data-driven Planning via Imitation Learning](https://mlsanity.com/paper/1711.06391/) (2017)
- [RayNet: Learning Volumetric 3D Reconstruction with Ray Potentials](https://mlsanity.com/paper/1901.01535/) (2019)
- [Differentiable Physics-informed Graph Networks](https://mlsanity.com/paper/1902.02950/) (2019)
- [Differentiable Algorithm Networks for Composable Robot Learning](https://mlsanity.com/paper/1905.11602/) (2019)
- [An Inductive Bias for Distances: Neural Nets that Respect the Triangle Inequality](https://mlsanity.com/paper/2002.05825/) (2020)
- [Learning Deep Structured Models](https://mlsanity.com/paper/1407.2538/) (2014)
- [Infinite-Horizon Differentiable Model Predictive Control](https://mlsanity.com/paper/2001.02244/) (2020)

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

- [Learning Graph Search Heuristics](https://mlsanity.com/paper/2212.03978/) (2022)
- [Learning Admissible Heuristics for A*: Theory and Practice](https://mlsanity.com/paper/2509.22626/) (2025)
- [DAA*: Deep Angular A Star for Image-based Path Planning](https://mlsanity.com/paper/2507.09305/) (2025)

## Abstract (arXiv)

Recently, the trend of incorporating differentiable algorithms into deep learning architectures arose in machine learning research, as the fusion of neural layers and algorithmic layers has been beneficial for handling combinatorial data, such as shortest paths on graphs. Recent works related to data-driven planning aim at learning either cost functions or heuristic functions, but not both. We propose Neural Weighted A*, a differentiable anytime planner able to produce improved representations of planar maps as graph costs and heuristics. Training occurs end-to-end on raw images with direct supervision on planning examples, thanks to a differentiable A* solver integrated into the architecture. More importantly, the user can trade off planning accuracy for efficiency at run-time, using a single, real-valued parameter. The solution suboptimality is constrained within a linear bound equal to the optimal path cost multiplied by the tradeoff parameter. We experimentally show the validity of our claims by testing Neural Weighted A* against several baselines, introducing a novel, tile-based navigation dataset. We outperform similar architectures in planning accuracy and efficiency.