# A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks

arXiv: https://arxiv.org/abs/2007.07442 · HTML: https://mlsanity.com/paper/2007.07442/

## Key facts

- Cited by 87 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jul 2020.

## Built on by (most cited first, 30 of 87)

- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [PirateNets: Physics-informed Deep Learning with Residual Adaptive Networks](https://mlsanity.com/paper/2402.00326/) (2024)
- [Separable Physics-Informed Neural Networks](https://mlsanity.com/paper/2306.15969/) (2023)
- [Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective](https://mlsanity.com/paper/2502.00604/) (2025)
- [A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions](https://mlsanity.com/paper/2302.14227/) (2023)
- [Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2501.16371/) (2025)
- [Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?](https://mlsanity.com/paper/2405.04230/) (2024)
- [On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elements](https://mlsanity.com/paper/2110.14121/) (2021)
- [Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks](https://mlsanity.com/paper/2205.02902/) (2022)
- [An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensions](https://mlsanity.com/paper/2309.07049/) (2023)
- [Improved Training of Physics-Informed Neural Networks with Model Ensembles](https://mlsanity.com/paper/2204.05108/) (2022)
- [A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations](https://mlsanity.com/paper/2101.01708/) (2021)
- [Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networks](https://mlsanity.com/paper/2310.04867/) (2023)
- [A Modified Batch Intrinsic Plasticity Method for Pre-training the Random Coefficients of Extreme Learning Machines](https://mlsanity.com/paper/2103.08042/) (2021)
- [Simulating Three-dimensional Turbulence with Physics-informed Neural Networks](https://mlsanity.com/paper/2507.08972/) (2025)
- [Random Weight Factorization Improves the Training of Continuous Neural Representations](https://mlsanity.com/paper/2210.01274/) (2022)
- [Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networks](https://mlsanity.com/paper/2201.09989/) (2022)
- [Numerical Computation of Partial Differential Equations by Hidden-Layer Concatenated Extreme Learning Machine](https://mlsanity.com/paper/2204.11375/) (2022)
- [Improving physics-informed DeepONets with hard constraints](https://mlsanity.com/paper/2309.07899/) (2023)
- [Exact Enforcement of Temporal Continuity in Sequential Physics-Informed Neural Networks](https://mlsanity.com/paper/2403.03223/) (2024)
- [Parametric Encoding with Attention and Convolution Mitigate Spectral Bias of Neural Partial Differential Equation Solvers](https://mlsanity.com/paper/2403.15652/) (2024)
- [Neural Networks Based on Power Method and Inverse Power Method for Solving Linear Eigenvalue Problems](https://mlsanity.com/paper/2209.11134/) (2022)
- [Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNs](https://mlsanity.com/paper/2210.03426/) (2022)

## Abstract (arXiv)

We present a simple and effective method for representing periodic functions and enforcing exactly the periodic boundary conditions for solving differential equations with deep neural networks (DNN). The method stems from some simple properties about function compositions involving periodic functions. It essentially composes a DNN-represented arbitrary function with a set of independent periodic functions with adjustable (training) parameters. We distinguish two types of periodic conditions: those imposing the periodicity requirement on the function and all its derivatives (to infinite order), and those imposing periodicity on the function and its derivatives up to a finite order k (kgeqslant 0). The former will be referred to as C∞ periodic conditions, and the latter Ck periodic conditions. We define operations that constitute a C∞ periodic layer and a Ck periodic layer (for any kgeqslant 0). A deep neural network with a C∞ (or Ck) periodic layer incorporated as the second layer automatically and exactly satisfies the C∞ (or Ck) periodic conditions. We present extensive numerical experiments on ordinary and partial differential equations with C∞ and Ck periodic boundary conditions to verify and demonstrate that the proposed method indeed enforces exactly, to the machine accuracy, the periodicity for the DNN solution and its derivatives.