# PINNup: Robust neural network wavefield solutions using frequency upscaling and neuron splitting

arXiv: https://arxiv.org/abs/2109.14536 · HTML: https://mlsanity.com/paper/2109.14536/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 17 references that are in the map.
- First posted on arXiv 29 Sep 2021.

## Builds on (most cited first, 4 of 4)

- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Net2Net: Accelerating Learning via Knowledge Transfer](https://mlsanity.com/paper/1511.05641/) (2015)
- [Splitting Steepest Descent for Growing Neural Architectures](https://mlsanity.com/paper/1910.02366/) (2019)

## Built on by (most cited first, 20 of 20)

- [Implicit Full Waveform Inversion with Deep Neural Representation](https://mlsanity.com/paper/2209.03525/) (2022)
- [Incremental Spatial and Spectral Learning of Neural Operators for Solving Large-Scale PDEs](https://mlsanity.com/paper/2211.15188/) (2022)
- [Efficient physics-informed neural networks using hash encoding](https://mlsanity.com/paper/2302.13397/) (2023)
- [InRank: Incremental Low-Rank Learning](https://mlsanity.com/paper/2306.11250/) (2023)
- [GaborPINN: Efficient physics informed neural networks using multiplicative filtered networks](https://mlsanity.com/paper/2308.05843/) (2023)
- [Microseismic source imaging using physics-informed neural networks with hard constraints](https://mlsanity.com/paper/2304.04315/) (2023)
- [LatentPINNs: Generative physics-informed neural networks via a latent representation learning](https://mlsanity.com/paper/2305.07671/) (2023)
- [Physics-informed neural wavefields with Gabor basis functions](https://mlsanity.com/paper/2310.10602/) (2023)
- [Physics-informed waveform inversion using pretrained wavefield neural operators](https://mlsanity.com/paper/2509.08967/) (2025)
- [RED-DiffEq: Regularization by denoising diffusion models for solving inverse PDE problems with application to full waveform inversion](https://mlsanity.com/paper/2509.21659/) (2025)
- [A Critical Review of Physics-Informed Machine Learning Applications in Subsurface Energy Systems](https://mlsanity.com/paper/2308.04457/) (2023)
- [Neuro-Spectral Architectures for Causal Physics-Informed Networks](https://mlsanity.com/paper/2509.04966/) (2025)
- [Physics-informed neural network for seismic wave inversion in layered semi-infinite domain](https://mlsanity.com/paper/2305.05150/) (2023)
- [An effective physics-informed neural operator framework for predicting wavefields](https://mlsanity.com/paper/2507.16431/) (2025)
- [PINNslope: seismic data interpolation and local slope estimation with physics informed neural networks](https://mlsanity.com/paper/2305.15990/) (2023)
- [PreAdaptFWI: Pretrained-Based Adaptive Residual Learning for Full-Waveform Inversion Without Dataset Dependency](https://mlsanity.com/paper/2502.11913/) (2025)
- [DiffPINN: Generative diffusion-initialized physics-informed neural networks for accelerating seismic wavefield representation](https://mlsanity.com/paper/2506.00471/) (2025)
- [Enforcing Reciprocity in Operator Learning for Seismic Wave Propagation](https://mlsanity.com/paper/2602.11631/) (2026)
- [Overview of Physics-Informed Machine Learning Inversion of Geophysical Data](https://mlsanity.com/paper/2310.08109/) (2023)
- [Staged Depth Training: A Representation Curriculum for PINNs](https://mlsanity.com/paper/2609.30299/) (2026)

## Abstract (arXiv)

Solving for the frequency-domain scattered wavefield via physics-informed neural network (PINN) has great potential in seismic modeling and inversion. However, when dealing with high-frequency wavefields, its accuracy and training cost limits its applications. Thus, we propose a novel implementation of PINN using frequency upscaling and neuron splitting, which allows the neural network model to grow in size as we increase the frequency while leveraging the information from the pre-trained model for lower-frequency wavefields, resulting in fast convergence to high-accuracy solutions. Numerical results show that, compared to the commonly used PINN with random initialization, the proposed PINN exhibits notable superiority in terms of convergence and accuracy and can achieve neuron based high-frequency wavefield solutions with a two-hidden-layer model.