Bayesian Physics Informed Neural Network

• BPINN is a framework that integrates Bayesian inference with physics-informed neural networks to enforce PDE constraints and quantify both epistemic and aleatory uncertainty.
• It employs methodologies such as variational inference and MCMC to derive a posterior distribution over neural network parameters, ensuring robust uncertainty quantification.
• BPINNs have been applied in solving forward PDEs, inverse problems, and system identification, offering scalable, mesh-free solutions in scientific computing.

Bayesian Physics Informed Neural Network (BPINN) is a framework that integrates Bayesian inference with physics-informed neural networks (PINNs), enabling the quantification of uncertainty in the solution of forward and inverse problems governed by partial differential equations (PDEs) or other physical laws. In BPINN, the neural network is trained not only to fit observed data but to enforce known physical constraints rigorously, while the Bayesian formalism provides a principled mechanism to handle epistemic and aleatory uncertainty, propagate errors, and estimate credible intervals for predictions and inferred parameters.

1. Theoretical Foundations

Bayesian Physics Informed Neural Networks combine two major methodological advances: PINNs and Bayesian neural networks (BNNs). PINNs encode physical laws, typically PDEs, by augmenting the loss function with terms penalizing residuals of the governing equations. Bayesian neural networks, in contrast to frequentist approaches, regard the weights and biases (or sometimes hyperparameters or outputs) of the network as random variables with prior distributions; training thus yields a posterior distribution over the network’s parameters conditioned on the data and (in the case of BPINN) physical constraints.

The typical BPINN approach starts with the Bayesian formulation: $$p(\theta \mid \mathcal{D}, \mathcal{P}) \propto p(\mathcal{D} \mid \theta) p(\mathcal{P} \mid \theta) p(\theta)$$ where $\theta$ represents the neural network parameters, $\mathcal{D}$ is observed data, $\mathcal{P}$ encodes the physics (e.g., PDE constraint residuals), and $p(\theta)$ is the prior. The likelihood $p(\mathcal{D} \mid \theta)$ enforces data fidelity, while $p(\mathcal{P} \mid \theta)$—often implemented via the exponential of the negative physics residual—ensures adherence to physical laws. Bayesian inference (via, e.g., variational inference or MCMC) yields not a point estimate but a full posterior over $\theta$.

2. Mathematical Formulation and Inference Procedures

A prototypical BPINN forwards the probabilistic PINN ansatz: Let $f_\theta(\boldsymbol{x})$ denote the neural network approximation of the physical field (e.g., solution $u(\boldsymbol{x})$), parameterized by random weights $\theta$. The data loss is typically composed as: $$\mathcal{L}_{\text{data}}(\theta) = \sum_{i=1}^{N_u} \left| f_\theta(\boldsymbol{x}_u^{(i)}) - u^{(i)} \right|^2$$ whereas the physics loss involves the squared residuals of the governing PDE at a collection of collocation points $\boldsymbol{x}_f$: $$\mathcal{L}_{\text{phys}}(\theta) = \sum_{j=1}^{N_f} \left| \mathcal{N}[f_\theta](\boldsymbol{x}_f^{(j)}) - s^{(j)} \right|^2$$ with $\mathcal{N}[\cdot]$ being a differential operator and $s$ source or forcing terms.

The Bayesian posterior over network parameters is: $$p(\theta \mid \mathcal{D}, \mathcal{P}) \propto \exp\left(-\frac{\mathcal{L}_{\text{data}}(\theta)}{2\sigma_u^2}\right) \exp\left(-\frac{\mathcal{L}_{\text{phys}}(\theta)}{2\sigma_f^2}\right) p(\theta)$$ where $\sigma_u$ and $\sigma_f$ are noise (or weighting) hyperparameters for data and physics constraints, respectively.

Posterior inference can be performed by:

• Stochastic Variational Inference (SVI): Approximating the posterior with a tractable variational family, optimizing ELBO.
• Markov Chain Monte Carlo (MCMC): Sampling from the true posterior, often computationally intensive for large networks.

The output is a predictive posterior: $$p(u^\ast \mid \boldsymbol{x}^\ast, \mathcal{D}, \mathcal{P}) = \int p(u^\ast \mid \boldsymbol{x}^\ast, \theta) p(\theta \mid \mathcal{D}, \mathcal{P}) d\theta$$ yielding both mean and credible interval for predictions at new inputs.

3. Uncertainty Quantification and Interpretability

Unlike deterministic PINNs, BPINNs natively quantify uncertainty in their predictions and learnt parameters. This uncertainty arises from two principal sources:

• Aleatory uncertainty: Inherent stochasticity of physical processes, accommodated via likelihood modeling.
• Epistemic uncertainty: Arising from finite data or incomplete knowledge, captured via the posterior over parameters.

The output variance enables assessment of prediction reliability, credible intervals for quantities of interest, and principled model selection (e.g., via marginal likelihood or Bayes factor). This is critical in safety-critical scientific and engineering applications, inverse design, and experiment planning.

Credible intervals are typically reported for physical observables, inferred boundary or material parameters, or places where extrapolation occurs (e.g., in the vicinity of unobserved domains or at future time steps).

4. Application Scenarios

BPINNs have been demonstrated in:

• Forward PDE solution under uncertainty in parameters or boundary conditions.
• Inverse problems: Inferring unknown coefficients, sources, or system parameters from partial/noisy observations.
• System identification where governing physics is only partially known.
• Multiphysics and multiscale coupling with heterogeneous available data.
• Robust design, uncertainty propagation, and experimental design leveraging quantified uncertainty.

In practice, BPINNs have been applied to canonical PDEs (Poisson, advection-diffusion, Navier-Stokes), ODEs, and even coupled systems where only some operators are known or only limited data is available.

5. Connections to Related Methods

BPINNs generalize and unify several threads in scientific machine learning:

• PINNs: Deterministic constraint enforcement, typically via $L_2$ penalization; does not yield uncertainty quantification.
• Bayesian Neural Networks: Uncertainty quantification, but no incorporation of physical constraints.
• Gaussian Processes (GPs): Closed-form uncertainty, but with scalability and flexibility limitations for high-dimensional or nonlinear PDEs; physics-informed GPs exist, but are often less scalable than BPINNs.
• Ensemble PINNs or MC Dropout PINNs: Ad hoc uncertainty estimation, not grounded in rigorous Bayesian inference.
• Bayesian Inverse Problems: Classical Bayesian inversion, but can be limited by computational burdens for complex parametric models, whereas BPINNs offer mesh-free, highly flexible function priors and scalable variational methods.

BPINNs represent an overview that retains the rigorous constraint satisfaction and expressiveness of PINNs, while adopting Bayesian statistical consistency and learning-calibrated uncertainty.

6. Computational Considerations and Challenges

The primary challenge in BPINNs is the computational burden of posterior inference in high-dimensional, non-convex neural parameter spaces. Variational inference offers scalability, but may underestimate uncertainty due to a restricted variational family; MCMC is more accurate but substantially less scalable. Hybrid or amortized inference strategies, active learning, and sparse neural architectures are active areas of methodology development.

Choice of noise-model hyperparameters $(\sigma_u, \sigma_f)$ dramatically impacts uncertainty calibration and can be learned as part of the Bayesian process, typically via a hierarchical prior.

Adaptive collocation strategies and multi-fidelity frameworks are often combined with BPINN to focus computational effort where uncertainty or residual is largest.

7. Current Research Frontiers

BPINNs remain at the cutting edge of uncertainty-aware scientific machine learning. Current research themes include:

• Adaptive refinement in space-time based on posterior-driven error indicators.
• Incorporation of additional sources of prior knowledge or non-Gaussian likelihood noise models.
• Extension to operator learning frameworks (e.g., DeepONet, FNO) with Bayesian variants.
• Automated selection of data versus physics weighting parameters.
• Rigorous benchmarking on high-dimensional and multi-scale PDE systems, including stochastic PDEs.

BPINNs are positioned as foundational methodology for robust AI-accelerated discovery, inverse modeling, and UQ in computational physics, with broad applicability from fluid dynamics to geosciences and biomedical engineering.