# Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systems

arXiv: https://arxiv.org/abs/2205.08304 · HTML: https://mlsanity.com/paper/2205.08304/

## Key facts

- Cited by 54 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 37 references that are in the map.
- First posted on arXiv 12 May 2022.

## Builds on (most cited first, 7 of 7)

- [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)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Multi-fidelity Bayesian Neural Networks: Algorithms and Applications](https://mlsanity.com/paper/2012.13294/) (2020)
- [GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systems](https://mlsanity.com/paper/2109.00092/) (2021)
- [Learning Functional Priors and Posteriors from Data and Physics](https://mlsanity.com/paper/2106.05863/) (2021)

## Built on by (most cited first, 30 of 54)

- [A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks](https://mlsanity.com/paper/2406.02917/) (2024)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [A new family of Constitutive Artificial Neural Networks towards automated model discovery](https://mlsanity.com/paper/2210.02202/) (2022)
- [Correcting model misspecification in physics-informed neural networks (PINNs)](https://mlsanity.com/paper/2310.10776/) (2023)
- [L-HYDRA: Multi-Head Physics-Informed Neural Networks](https://mlsanity.com/paper/2301.02152/) (2023)
- [Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operators](https://mlsanity.com/paper/2311.11262/) (2023)
- [Discovering a reaction-diffusion model for Alzheimer's disease by combining PINNs with symbolic regression](https://mlsanity.com/paper/2307.08107/) (2023)
- [Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemble](https://mlsanity.com/paper/2503.06320/) (2025)
- [Leveraging viscous Hamilton-Jacobi PDEs for uncertainty quantification in scientific machine learning](https://mlsanity.com/paper/2404.08809/) (2024)
- [Initialization-enhanced Physics-Informed Neural Network with Domain Decomposition (IDPINN)](https://mlsanity.com/paper/2406.03172/) (2024)
- [A Generative Modeling Framework for Inferring Families of Biomechanical Constitutive Laws in Data-Sparse Regimes](https://mlsanity.com/paper/2305.03184/) (2023)
- [Fast meta-solvers for 3D complex-shape scatterers using neural operators trained on a non-scattering problem](https://mlsanity.com/paper/2405.12380/) (2024)
- [Approaching epidemiological dynamics of COVID-19 with physics-informed neural networks](https://mlsanity.com/paper/2302.08796/) (2023)
- [Physics-informed PointNet: On how many irregular geometries can it solve an inverse problem simultaneously? Application to linear elasticity](https://mlsanity.com/paper/2303.13634/) (2023)
- [Randomized Physics-Informed Neural Networks for Bayesian Data Assimilation](https://mlsanity.com/paper/2407.04617/) (2024)
- [Accounting for plasticity: An extension of inelastic Constitutive Artificial Neural Networks](https://mlsanity.com/paper/2407.19326/) (2024)
- [NeuroSEM: A hybrid framework for simulating multiphysics problems by coupling PINNs and spectral elements](https://mlsanity.com/paper/2407.21217/) (2024)
- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [LatentPINNs: Generative physics-informed neural networks via a latent representation learning](https://mlsanity.com/paper/2305.07671/) (2023)
- [A Framework for Strategic Discovery of Credible Neural Network Surrogate Models under Uncertainty](https://mlsanity.com/paper/2403.08901/) (2024)
- [Uncertainty Quantification in PINNs for Turbulent Flows: Bayesian Inference and Repulsive Ensembles](https://mlsanity.com/paper/2604.17156/) (2026)
- [WANCO: Weak Adversarial Networks for Constrained Optimization problems](https://mlsanity.com/paper/2407.03647/) (2024)
- [Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiology](https://mlsanity.com/paper/2408.07201/) (2024)
- [Physics-informed KAN PointNet: Deep learning for simultaneous solutions to inverse problems in incompressible flow on numerous irregular geometries](https://mlsanity.com/paper/2504.06327/) (2025)
- [Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundles](https://mlsanity.com/paper/2505.06459/) (2025)
- [Bayesian Physics Informed Neural Networks for Reliable Transformer Prognostics](https://mlsanity.com/paper/2509.15933/) (2025)
- [Bayesian identification of nonseparable Hamiltonians with multiplicative noise using deep learning and reduced-order modeling](https://mlsanity.com/paper/2401.12476/) (2024)
- [Enhanced BPINN Training Convergence in Solving General and Multi-scale Elliptic PDEs with Noise](https://mlsanity.com/paper/2408.09340/) (2024)
- [Empowering Bayesian Neural Networks with Functional Priors through Anchored Ensembling for Mechanics Surrogate Modeling Applications](https://mlsanity.com/paper/2409.05234/) (2024)
- [Physics-informed deep learning for infectious disease forecasting](https://mlsanity.com/paper/2501.09298/) (2025)

## Abstract (arXiv)

Understanding real-world dynamical phenomena remains a challenging task. Across various scientific disciplines, machine learning has advanced as the go-to technology to analyze nonlinear dynamical systems, identify patterns in big data, and make decision around them. Neural networks are now consistently used as universal function approximators for data with underlying mechanisms that are incompletely understood or exceedingly complex. However, neural networks alone ignore the fundamental laws of physics and often fail to make plausible predictions. Here we integrate data, physics, and uncertainties by combining neural networks, physics-informed modeling, and Bayesian inference to improve the predictive potential of traditional neural network models. We embed the physical model of a damped harmonic oscillator into a fully-connected feed-forward neural network to explore a simple and illustrative model system, the outbreak dynamics of COVID-19. Our Physics-Informed Neural Networks can seamlessly integrate data and physics, robustly solve forward and inverse problems, and perform well for both interpolation and extrapolation, even for a small amount of noisy and incomplete data. At only minor additional cost, they can self-adaptively learn the weighting between data and physics. Combined with Bayesian Neural Networks, they can serve as priors in a Bayesian Inference, and provide credible intervals for uncertainty quantification. Our study reveals the inherent advantages and disadvantages of Neural Networks, Bayesian Inference, and a combination of both and provides valuable guidelines for model selection. While we have only demonstrated these approaches for the simple model problem of a seasonal endemic infectious disease, we anticipate that the underlying concepts and trends generalize to more complex disease conditions and, more broadly, to a wide variety of nonlinear dynamical systems.