# Physics-informed neural network solves minimal surfaces in curved spacetime

arXiv: https://arxiv.org/abs/2509.10866 · HTML: https://mlsanity.com/paper/2509.10866/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 64 references that are in the map.
- First posted on arXiv 13 Sep 2025.

## Builds on (most cited first, 16 of 16)

- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [Physics informed deep learning for computational elastodynamics without labeled data](https://mlsanity.com/paper/2006.08472/) (2020)
- [Hamiltonian neural networks for solving equations of motion](https://mlsanity.com/paper/2001.11107/) (2020)
- [Neural Network Approximations for Calabi-Yau Metrics](https://mlsanity.com/paper/2012.15821/) (2020)
- [Learning Size and Shape of Calabi-Yau Spaces](https://mlsanity.com/paper/2111.01436/) (2021)
- [A cusp-capturing PINN for elliptic interface problems](https://mlsanity.com/paper/2210.08424/) (2022)
- [Unsupervised Neural Networks for Quantum Eigenvalue Problems](https://mlsanity.com/paper/2010.05075/) (2020)
- [IDRLnet: A Physics-Informed Neural Network Library](https://mlsanity.com/paper/2107.04320/) (2021)
- [Gravitational Duals from Equations of State](https://mlsanity.com/paper/2403.14763/) (2024)
- [AdS/Deep-Learning made easy: simple examples](https://mlsanity.com/paper/2011.13726/) (2020)
- [Machine-learning emergent spacetime from linear response in future tabletop quantum gravity experiments](https://mlsanity.com/paper/2411.16052/) (2024)
- [Machine Learning-Driven Optimization of TPMS Architected Materials Using Simulated Annealing](https://mlsanity.com/paper/2406.05142/) (2024)

## Built on by (most cited first, 6 of 6)

- [Aspects of holographic entanglement using physics-informed-neural-networks](https://mlsanity.com/paper/2509.25311/) (2025)
- [Minimising Willmore Energy via Neural Flow](https://mlsanity.com/paper/2604.04321/) (2026)
- [Minimal surfaces, Knots, and Neural Networks](https://mlsanity.com/paper/2605.26234/) (2026)
- [PINNs in More General Geometry](https://mlsanity.com/paper/2604.25020/) (2026)
- [Neural Representation of Minimal Surfaces](https://mlsanity.com/paper/2607.23437/) (2026)
- [A user's guide to PINNs in geometric analysis: lessons from the asymptotic Plateau problem](https://mlsanity.com/paper/2607.28733/) (2026)

## Abstract (arXiv)

We develop a flexible framework based on physics-informed neural networks (PINNs) for solving boundary value problems involving minimal surfaces in curved spacetimes, with a particular emphasis on singularities and moving boundaries. By encoding the underlying physical laws into the loss function and designing network architectures that incorporate the singular behavior and dynamic boundaries, our approach enables robust and accurate solutions to both ordinary and partial differential equations with complex boundary conditions. We demonstrate the versatility of this framework through applications to minimal surface problems in anti-de Sitter (AdS) spacetime, including examples relevant to the AdS/CFT correspondence (e.g. Wilson loops and gluon scattering amplitudes) popularly used in the context of string theory in theoretical physics. Our methods efficiently handle singularities at boundaries, and also support both "soft" (loss-based) and "hard" (formulation-based) imposition of boundary conditions, including cases where the position of a boundary is promoted to a trainable parameter. The techniques developed here are not limited to high-energy theoretical physics but are broadly applicable to boundary value problems encountered in mathematics, engineering, and the natural sciences, wherever singularities and moving boundaries play a critical role.