# PINN Training using Biobjective Optimization: The Trade-off between Data Loss and Residual Loss

arXiv: https://arxiv.org/abs/2302.01810 · HTML: https://mlsanity.com/paper/2302.01810/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 46 references that are in the map.
- First posted on arXiv 3 Feb 2023.

## Builds on (most cited first, 15 of 15)

- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Multi-Task Learning as Multi-Objective Optimization](https://mlsanity.com/paper/1810.04650/) (2018)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Meta-learning PINN loss functions](https://mlsanity.com/paper/2107.05544/) (2021)
- [The stochastic multi-gradient algorithm for multi-objective optimization and its application to supervised machine learning](https://mlsanity.com/paper/1907.04472/) (2019)
- [Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2107.00940/) (2021)
- [Data vs. Physics: The Apparent Pareto Front of Physics-Informed Neural Networks](https://mlsanity.com/paper/2105.00862/) (2021)
- [Data-driven approaches for predicting spread of infectious diseases through DINNs: Disease Informed Neural Networks](https://mlsanity.com/paper/2110.05445/) (2021)
- [Kalman-Bucy-Informed Neural Network for System Identification](https://mlsanity.com/paper/2210.03424/) (2022)
- [Efficient and Sparse Neural Networks by Pruning Weights in a Multiobjective Learning Approach](https://mlsanity.com/paper/2008.13590/) (2020)
- [A simple and efficient dichotomic search algorithm for multi-objective mixed integer linear programs](https://mlsanity.com/paper/1911.08937/) (2019)

## Built on by (most cited first, 6 of 6)

- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunities](https://mlsanity.com/paper/2501.06572/) (2025)
- [PINNverse: Accurate parameter estimation in differential equations from noisy data with constrained physics-informed neural networks](https://mlsanity.com/paper/2504.05248/) (2025)
- [Common pitfalls to avoid while using multiobjective optimization in machine learning](https://mlsanity.com/paper/2405.01480/) (2024)
- [Identifying Memory Effects in Epidemics via a Fractional SEIRD Model and Physics-Informed Neural Networks](https://mlsanity.com/paper/2509.22760/) (2025)
- [Number Theoretic Accelerated Learning of Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.13869/) (2023)

## Abstract (arXiv)

Physics informed neural networks (PINNs) have proven to be an efficient tool to represent problems for which measured data are available and for which the dynamics in the data are expected to follow some physical laws. In this paper, we suggest a multiobjective perspective on the training of PINNs by treating the data loss and the residual loss as two individual objective functions in a truly biobjective optimization approach. As a showcase example, we consider COVID-19 predictions in Germany and built an extended susceptibles-infected-recovered (SIR) model with additionally considered leaky-vaccinated and hospitalized populations (SVIHR model) to model the transition rates and to predict future infections. SIR-type models are expressed by systems of ordinary differential equations (ODEs). We investigate the suitability of the generated PINN for COVID-19 predictions and compare the resulting predicted curves with those obtained by applying the method of non-standard finite differences to the system of ODEs and initial data. The approach is applicable to various systems of ODEs that define dynamical regimes. Those regimes do not need to be SIR-type models, and the corresponding underlying data sets do not have to be associated with COVID-19.