# Compliance Minimization via Physics-Informed Gaussian Processes

arXiv: https://arxiv.org/abs/2507.09968 · HTML: https://mlsanity.com/paper/2507.09968/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 53 references that are in the map.
- First posted on arXiv 14 Jul 2025.

## Builds on (most cited first, 21 of 21)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Curriculum Learning: A Survey](https://mlsanity.com/paper/2101.10382/) (2021)
- [Physics informed deep learning for computational elastodynamics without labeled data](https://mlsanity.com/paper/2006.08472/) (2020)
- [TopologyGAN: Topology Optimization Using Generative Adversarial Networks Based on Physical Fields Over the Initial Domain](https://mlsanity.com/paper/2003.04685/) (2020)
- [Neural reparameterization improves structural optimization](https://mlsanity.com/paper/1909.04240/) (2019)
- [NTopo: Mesh-free Topology Optimization using Implicit Neural Representations](https://mlsanity.com/paper/2102.10782/) (2021)
- [The mixed deep energy method for resolving concentration features in finite strain hyperelasticity](https://mlsanity.com/paper/2104.09623/) (2021)
- [Topology Optimization via Machine Learning and Deep Learning: A Review](https://mlsanity.com/paper/2210.10782/) (2022)
- [PIXEL: Physics-Informed Cell Representations for Fast and Accurate PDE Solvers](https://mlsanity.com/paper/2207.12800/) (2022)
- [Calibrating constitutive models with full-field data via physics informed neural networks](https://mlsanity.com/paper/2203.16577/) (2022)
- [A deep learning energy-based method for classical elastoplasticity](https://mlsanity.com/paper/2209.06467/) (2022)
- [Neural topology optimization: the good, the bad, and the ugly](https://mlsanity.com/paper/2407.13954/) (2024)
- [Parametric Encoding with Attention and Convolution Mitigate Spectral Bias of Neural Partial Differential Equation Solvers](https://mlsanity.com/paper/2403.15652/) (2024)
- [A Gaussian Process Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2401.03492/) (2024)
- [Data-driven topology design using a deep generative model](https://mlsanity.com/paper/2006.04559/) (2020)
- [Deep energy method in topology optimization applications](https://mlsanity.com/paper/2207.03072/) (2022)
- [DMF-TONN: Direct Mesh-free Topology Optimization using Neural Networks](https://mlsanity.com/paper/2305.04107/) (2023)
- [A new generation 99 line Matlab code for compliance Topology Optimization and its extension to 3D](https://mlsanity.com/paper/2005.05436/) (2020)
- [Simultaneous and Meshfree Topology Optimization with Physics-informed Gaussian Processes](https://mlsanity.com/paper/2408.03490/) (2024)
- [Dynamically configured physics-informed neural network in topology optimization applications](https://mlsanity.com/paper/2312.06993/) (2023)
- [Localized Physics-informed Gaussian Processes with Curriculum Training for Topology Optimization](https://mlsanity.com/paper/2503.15561/) (2025)

## Built on by (most cited first, 2 of 2)

- [Multi-material Multi-physics Topology Optimization with Physics-informed Gaussian Process Priors](https://mlsanity.com/paper/2602.17783/) (2026)
- [On the Importance of Geometric Nonlinearity and Temperature-Dependent Properties in Multi-Material Thermo-Mechanical Topology Optimization](https://mlsanity.com/paper/2608.10344/) (2026)

## Abstract (arXiv)

Machine learning (ML) techniques have recently gained significant attention for solving compliance minimization (CM) problems. However, these methods typically provide poor feature boundaries, are very expensive, and lack a systematic mechanism to control the design complexity. Herein, we address these limitations by proposing a mesh-free and simultaneous framework based on physics-informed Gaussian processes (GPs). In our approach, we parameterize the design and state variables with GP priors which have independent kernels but share a multi-output neural network (NN) as their mean function. The architecture of this NN is based on Parametric Grid Convolutional Attention Networks (PGCANs) which not only mitigate spectral bias issues, but also provide an interpretable mechanism to control design complexity. We estimate all the parameters of our GP-based representations by simultaneously minimizing the compliance, total potential energy, and residual of volume fraction constraint. Importantly, our loss function exclude all data-based residuals as GPs automatically satisfy them. We also develop computational schemes based on curriculum training and numerical integration to increase the efficiency and robustness of our approach which is shown to (1) produce super-resolution topologies with fast convergence, (2) achieve comparable compliance and less gray area fraction compared to traditional numerical methods, (3) provide control over fine-scale features, and (4) outperform competing ML-based methods.