# How to Avoid Trivial Solutions in Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2112.05620 · HTML: https://mlsanity.com/paper/2112.05620/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 17 references that are in the map.
- First posted on arXiv 10 Dec 2021.

## Builds on (most cited first, 5 of 5)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [CFDNet: a deep learning-based accelerator for fluid simulations](https://mlsanity.com/paper/2005.04485/) (2020)

## Built on by (most cited first, 15 of 15)

- [On the Role of Fixed Points of Dynamical Systems in Training Physics-Informed Neural Networks](https://mlsanity.com/paper/2203.13648/) (2022)
- [Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEs](https://mlsanity.com/paper/2403.10642/) (2024)
- [Physical Activation Functions (PAFs): An Approach for More Efficient Induction of Physics into Physics-Informed Neural Networks (PINNs)](https://mlsanity.com/paper/2205.14630/) (2022)
- [Generating Physical Dynamics under Priors](https://mlsanity.com/paper/2409.00730/) (2024)
- [Enabling Automatic Differentiation with Mollified Graph Neural Operators](https://mlsanity.com/paper/2504.08277/) (2025)
- [Projected Neural Differential Equations for Learning Constrained Dynamics](https://mlsanity.com/paper/2410.23667/) (2024)
- [Physics-aware deep learning framework for linear elasticity](https://mlsanity.com/paper/2302.09668/) (2023)
- [SetPINNs: Set-based Physics-informed Neural Networks](https://mlsanity.com/paper/2409.20206/) (2024)
- [Dynamic Reconstruction of Ultrasound-Derived Flow Fields With Physics-Informed Neural Fields](https://mlsanity.com/paper/2511.01804/) (2025)
- [Per-Loss Adapters for Gradient Conflict in Physics-Informed Neural Networks](https://mlsanity.com/paper/2605.10136/) (2026)
- [An Operator Learning Framework for Spatiotemporal Super-resolution of Scientific Simulations](https://mlsanity.com/paper/2311.02328/) (2023)
- [Transport-Embedded Neural Architecture: Redefining the Landscape of physics aware neural models in fluid mechanics](https://mlsanity.com/paper/2410.04114/) (2024)
- [B-PL-PINN: Stabilizing PINN Training with Bayesian Pseudo Labeling](https://mlsanity.com/paper/2507.01714/) (2025)
- [IP-Basis PINNs: Efficient Multi-Query Inverse Parameter Estimation](https://mlsanity.com/paper/2509.07245/) (2025)
- [Physics-informed Reinforcement Learning for Stochastic Reach-Avoid Analysis](https://mlsanity.com/paper/2608.17117/) (2026)

## Abstract (arXiv)

The advent of scientific machine learning (SciML) has opened up a new field with many promises and challenges in the field of simulation science by developing approaches at the interface of physics- and data-based modelling. To this end, physics-informed neural networks (PINNs) have been introduced in recent years, which cope for the scarcity in training data by incorporating physics knowledge of the problem at so-called collocation points. In this work, we investigate the prediction performance of PINNs with respect to the number of collocation points used to enforce the physics-based penalty terms. We show that PINNs can fail, learning a trivial solution that fulfills the physics-derived penalty term by definition. We have developed an alternative sampling approach and a new penalty term enabling us to remedy this core problem of PINNs in data-scarce settings with competitive results while reducing the amount of collocation points needed by up to 80 % for benchmark problems.