# Eco-PiNN: A Physics-informed Neural Network for Eco-toll Estimation

arXiv: https://arxiv.org/abs/2301.05739 · HTML: https://mlsanity.com/paper/2301.05739/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 28 references that are in the map.
- First posted on arXiv 13 Jan 2023.

## Builds on (most cited first, 7 of 7)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016)
- [A Survey on Multi-Task Learning](https://mlsanity.com/paper/1707.08114/) (2017)
- [Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems](https://mlsanity.com/paper/2003.04919/) (2020)
- [A Theoretically Grounded Application of Dropout in Recurrent Neural Networks](https://mlsanity.com/paper/1512.05287/) (2015)
- [ETA Prediction with Graph Neural Networks in Google Maps](https://mlsanity.com/paper/2108.11482/) (2021)

## Built on by (most cited first, 4 of 4)

- [Reducing Uncertainty in Sea-level Rise Prediction: A Spatial-variability-aware Approach](https://mlsanity.com/paper/2310.15179/) (2023)
- [Towards Physics-informed Diffusion for Anomaly Detection in Trajectories](https://mlsanity.com/paper/2506.06999/) (2025)
- [Towards Physics-Guided Foundation Models](https://mlsanity.com/paper/2502.15013/) (2025)
- [Towards Fine-Tuning-Based Site Calibration for Knowledge-Guided Machine Learning: A Summary of Results](https://mlsanity.com/paper/2512.16013/) (2025)

## Abstract (arXiv)

The eco-toll estimation problem quantifies the expected environmental cost (e.g., energy consumption, exhaust emissions) for a vehicle to travel along a path. This problem is important for societal applications such as eco-routing, which aims to find paths with the lowest exhaust emissions or energy need. The challenges of this problem are three-fold: (1) the dependence of a vehicle's eco-toll on its physical parameters; (2) the lack of access to data with eco-toll information; and (3) the influence of contextual information (i.e. the connections of adjacent segments in the path) on the eco-toll of road segments. Prior work on eco-toll estimation has mostly relied on pure data-driven approaches and has high estimation errors given the limited training data. To address these limitations, we propose a novel Eco-toll estimation Physics-informed Neural Network framework (Eco-PiNN) using three novel ideas, namely, (1) a physics-informed decoder that integrates the physical laws of the vehicle engine into the network, (2) an attention-based contextual information encoder, and (3) a physics-informed regularization to reduce overfitting. Experiments on real-world heavy-duty truck data show that the proposed method can greatly improve the accuracy of eco-toll estimation compared with state-of-the-art methods.