# Neural Fields with Hard Constraints of Arbitrary Differential Order

arXiv: https://arxiv.org/abs/2306.08943 · HTML: https://mlsanity.com/paper/2306.08943/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 42 references that are in the map.
- First posted on arXiv 15 Jun 2023.

## Builds on (most cited first, 20 of 20)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [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)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [SAL: Sign Agnostic Learning of Shapes from Raw Data](https://mlsanity.com/paper/1911.10414/) (2019)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [NeX: Real-time View Synthesis with Neural Basis Expansion](https://mlsanity.com/paper/2103.05606/) (2021)
- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [Implicit Deep Learning](https://mlsanity.com/paper/1908.06315/) (2019)
- [Neural Fields as Learnable Kernels for 3D Reconstruction](https://mlsanity.com/paper/2111.13674/) (2021)
- [Factor Fields: A Unified Framework for Neural Fields and Beyond](https://mlsanity.com/paper/2302.01226/) (2023)
- [Learning differentiable solvers for systems with hard constraints](https://mlsanity.com/paper/2207.08675/) (2022)
- [Neural BRDF Representation and Importance Sampling](https://mlsanity.com/paper/2102.05963/) (2021)
- [POLICE: Provably Optimal Linear Constraint Enforcement for Deep Neural Networks](https://mlsanity.com/paper/2211.01340/) (2022)
- [Linearly Constrained Neural Networks](https://mlsanity.com/paper/2002.01600/) (2020)
- [NeuMIP: Multi-Resolution Neural Materials](https://mlsanity.com/paper/2104.02789/) (2021)
- [Shape Transformation Using Variational Implicit Functions](https://mlsanity.com/paper/2303.02937/) (2023)

## Built on by (most cited first, 5 of 5)

- [Scaling physics-informed hard constraints with mixture-of-experts](https://mlsanity.com/paper/2402.13412/) (2024)
- [Robust Biharmonic Skinning Using Geometric Fields](https://mlsanity.com/paper/2406.00238/) (2024)
- [HoP: Homeomorphic Polar Learning for Hard Constrained Optimization](https://mlsanity.com/paper/2502.00304/) (2025)
- [mPOLICE: Provable Enforcement of Multi-Region Affine Constraints in Deep Neural Networks](https://mlsanity.com/paper/2502.02434/) (2025)
- [Learning-guided Kansa collocation for forward and inverse PDEs beyond linearity](https://mlsanity.com/paper/2602.07970/) (2026)

## Abstract (arXiv)

While deep learning techniques have become extremely popular for solving a broad range of optimization problems, methods to enforce hard constraints during optimization, particularly on deep neural networks, remain underdeveloped. Inspired by the rich literature on meshless interpolation and its extension to spectral collocation methods in scientific computing, we develop a series of approaches for enforcing hard constraints on neural fields, which we refer to as Constrained Neural Fields (CNF). The constraints can be specified as a linear operator applied to the neural field and its derivatives. We also design specific model representations and training strategies for problems where standard models may encounter difficulties, such as conditioning of the system, memory consumption, and capacity of the network when being constrained. Our approaches are demonstrated in a wide range of real-world applications. Additionally, we develop a framework that enables highly efficient model and constraint specification, which can be readily applied to any downstream task where hard constraints need to be explicitly satisfied during optimization.