# Finding geodesics with the Deep Ritz method

arXiv: https://arxiv.org/abs/2510.15177 · HTML: https://mlsanity.com/paper/2510.15177/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 38 references that are in the map.
- First posted on arXiv 16 Oct 2025.

## Builds on (most cited first, 15 of 15)

- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometries](https://mlsanity.com/paper/2004.06490/) (2020)
- [A deep learning energy method for hyperelasticity and viscoelasticity](https://mlsanity.com/paper/2201.08690/) (2022)
- [Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networks](https://mlsanity.com/paper/2308.12716/) (2023)
- [Feature-Based Interpolation and Geodesics in the Latent Spaces of Generative Models](https://mlsanity.com/paper/1904.03445/) (2019)
- [Atlas Generative Models and Geodesic Interpolation](https://mlsanity.com/paper/2102.00264/) (2021)
- [Einstein Fields: A Neural Perspective To Computational General Relativity](https://mlsanity.com/paper/2507.11589/) (2025)
- [Learning Geodesics of Geometric Shape Deformations From Images](https://mlsanity.com/paper/2410.18797/) (2024)
- [Variational volume reconstruction with the Deep Ritz Method](https://mlsanity.com/paper/2508.08309/) (2025)

## Built on by (most cited first, 1 of 1)

- [A Neural Network Framework for Geodesic-Like Curve Computation on Parametric Surfaces](https://mlsanity.com/paper/2606.18759/) (2026)

## Abstract (arXiv)

Geodesic problems involve computing trajectories between prescribed initial and final states to minimize a user-defined measure of distance, cost, or energy. They arise throughout physics and engineering -- for instance, in determining optimal paths through complex environments, modeling light propagation in refractive media, and the study of spacetime trajectories in control theory and general relativity. Despite their ubiquity, the scientific machine learning (SciML) community has given relatively little attention to investigating its methods in the context of these problems. In this work, we argue that given their simple geometry, variational structure, and natural nonlinearity, geodesic problems are particularly well-suited for the Deep Ritz method. We substantiate this claim with four numerical examples drawn from path planning, optics, solid mechanics, and generative modeling. Our goal is not to provide an exhaustive study of geodesic problems, but rather to identify a promising application of the Deep Ritz method and a fruitful direction for future SciML research.