Physics-Informed Bayesian Analysis

• Physics-informed Bayesian analyses are probabilistic methods that integrate physical laws, such as PDEs, with Bayesian inference using models like PI-GANs and PINNs.
• They utilize a two-stage framework that first learns a flexible functional prior from data and physics, then performs efficient posterior estimation in a reduced latent space via techniques like HMC.
• Applications include meta-learning, inverse problems, and high-dimensional regression, enabling robust uncertainty quantification and reliable predictions even under data scarcity.

Physics-informed Bayesian analyses refer to a class of probabilistic machine learning methods that integrate the governing laws of physical systems, often expressed as partial differential equations (PDEs) or other operator constraints, into a Bayesian inference framework. These approaches leverage both historical or sensor data and “hard” physical constraints to construct flexible, uncertainty-quantified models that enable robust extrapolation, meta-learning, and inference under data scarcity and noise. Central to this paradigm is the synthesis of deep generative modeling (such as physics-informed generative adversarial networks), operator learning, and advanced posterior estimation in reduced latent or parameter spaces.

1. Two-Stage Physics-Informed Bayesian Framework

The two-stage framework introduced in "Learning Functional Priors and Posteriors from Data and Physics" (Meng et al., 2021) comprises:

• Prior Learning Stage: A physics-informed generative model, specifically a PI-GAN (Physics-Informed Generative Adversarial Network), is trained using historical (meta-training) data and encoded physical laws to learn a flexible, functional prior over solution spaces. The generator network $G_\eta(x; \xi)$ receives both spatial/temporal coordinates $x$ and a latent code $\xi$, typically drawn from a standard normal distribution. This generator is tasked with synthesizing fields (e.g., $u(x)$), matching the distribution of historical or synthetic "snapshots" that incorporate solution fields, source terms, and boundary data. The GAN objective can adopt Wasserstein loss with a gradient penalty for stability and interpretability.

For example, the generator snapshots are $$Q_\eta(\xi) = (\tilde{u}_\eta(x_1; \xi), \tilde{u}_\eta(x_2; \xi), ...)$$ with GAN losses:

$$\begin{aligned} L_G &= - \mathbb{E}_{\xi \sim P(\xi)} [ D_\rho(Q_\eta(\xi)) ] \ L_D &= \mathbb{E}_{\xi \sim P(\xi)} [ D_\rho(Q_\eta(\xi)) ] - \mathbb{E}_{T \sim P_r} [ D_\rho(T) ] + \lambda \mathbb{E}_{\tilde{Z} \sim P_i} [ ( \|\nabla_{\tilde{Z}} D_\rho(\tilde{Z})\|_2 - 1 )^2 ], \end{aligned}$$

where $P_r$ is the real data distribution.

• Posterior Estimation Stage: With the PI-GAN generator fixed, Bayesian inference is performed in the low-dimensional latent space $\xi$ using new, typically sparse and noisy, observations $\mathcal{D}$. The posterior, $$P(\xi|\mathcal{D}) \propto P(\mathcal{D}|\xi)P(\xi)$$, is sampled using Hamiltonian Monte Carlo (HMC) or related schemes, and each posterior sample is mapped through $G_\eta$ to propagate uncertainty to field solutions, PDE coefficients, or source terms.

The data likelihood for fields, source, and boundaries is formally:

$$P(\mathcal{D}_u|\xi) = \prod_{i} \frac{1}{\sqrt{2\pi \sigma_i^2}} \exp \left\{ - \frac{ [\tilde{u}_\eta(x_i; \xi) - u_i]^2 }{2\sigma_i^2} \right\}$$

and the full posterior includes all observed quantities.

2. Physics Encoding: PINNs and Operator Learning

Physical constraints are incorporated using two complementary mechanisms:

• Physics-Informed Neural Networks (PINNs): For problems where the governing PDE is explicit, PI-GAN’s generator directly produces candidate fields (e.g., $u(x)$), and automatic differentiation enforces PDE and boundary residuals. For example, for $D \partial_x^2 u - k_r u^3 = f$ with Dirichlet BCs, the generator is differentiated to yield $\tilde{f}_\eta(x; \xi)$ and enforced against known $(x_i, f_i)$. This provides “hard” physics regularization during prior learning and shapes the likelihood in posterior sampling.
• Operator Regression via DeepONet: In scenarios where the explicit PDE is unknown or the system is complex (PDE-agnostic), a pre-trained DeepONet surrogate maps sources and boundary conditions to solution fields, $u = \tilde{S}[f, b](x)$. Independent generative models produce $f$ and $b$, and DeepONet composes these to generate solution samples. This operator-centric view decouples physical modeling from explicit PDE forms, enabling application to high-dimensional stochastic problems where direct physical model access is unavailable (e.g., fractional diffusion, stochastic porous media flow).

3. Scalability via Latent Space Inference and Posterior Estimation

A critical design choice is performing Bayesian inference in the low-dimensional latent space of the generator ($\dim(\xi) \sim 10\ldots100$). Rather than performing full posterior sampling in the high-dimensional weight space of typical Bayesian neural networks, the approach reduces computational complexity and variance in estimates, while maintaining expressiveness for functional uncertainty.

HMC (and extensions such as stochastic HMC or normalizing flows) is used for efficient sampling. The approach is amenable to mini-batch Monte Carlo inference for large-scale or “big data” problems, as the latent variable is directly conditioned on observed data, and computational cost is decoupled from the dimensionality of the function space.

4. Applications: Meta-Learning, Inverse Problems, High-Dimensional Regression

The framework is validated across diverse benchmark and real-world tasks:

• Meta-Learning: Applied to regression tasks with families of parameterized functions $A \cdot \sin(\omega x)$, the method leverages historical data to construct functional priors, yielding predictive uncertainty bounds that are narrower than those from standard GP regression or gradient-based meta-learners even with very limited new data.
• Forward and Inverse PDEs: PDE-constrained problems (e.g., nonlinear diffusion–reaction with unknown coefficients $k_r$ that may themselves be spatial fields) are solved by treating these coefficients as part of the generative model output, enabling both forward simulations and probabilistic inversion with uncertainty quantification.
• Spatio-Temporal and High-Dimensional Regression: When applied to real-world spatio-temporal data (e.g., marine riser displacement, 2D/100D stochastic flows), the approach accurately reconstructs complex fields from noisy, scattered measurements, with credible intervals on predictions matching empirical errors.

In all cases, the method demonstrates reliable posterior coverage: errors are typically bounded by two standard deviations from the predictive posterior, attesting to both the expressiveness of the prior and correctness of the posterior inference.

5. Numerical Performance, Extensions, and Trade-offs

The reduction in inference dimensionality confers substantial runtime and memory advantages, enabling sampling and uncertainty propagation for PDEs with up to 100 parametric dimensions. The latent space approach maintains both coverage and sharpness of posterior predictions, as the generator $G_\eta$ can be made deep and expressive without overwhelming posterior computation.

Potential extensions include leveraging stochastic HMC for mini-batch scalability, using normalizing flows to capture non-Gaussian posteriors, and operator-based surrogates for further efficiency. One empirical trade-off is that while PI-GANs can learn arbitrarily complex priors, training stability and data representativeness remain critical; care must be taken in GAN training design (e.g., regularization and sample diversity).

In scenarios where physics is only partially known, the DeepONet approach allows for “physics-aware” modeling without explicit PDE access, but its reliability will ultimately depend on the coverage and accuracy of operator regression.

6. Limitations and Practical Implementation Considerations

While the approach provides compelling uncertainty quantification and flexibility, several limitations exist:

• The quality and representativeness of the functional prior critically depend on the diversity and fidelity of historical or synthetic training data.
• The method requires specialized architectures (PI-GANs, DeepONet, and associated discriminators), and efficient HMC implementations in low-dimensional latent spaces.
• For very high-dimensional latent codes, normalizing flow parameterizations or stochastic HMC become necessary to preserve mixing and expressiveness.
• Careful physics encoding and validation (e.g., monitoring whether the generated fields satisfy hard constraints in the absence of direct differentiation) are essential for model reliability.
• While scalable, the method can be bottlenecked by GAN/discriminator training or computational operator surrogates in high-dimensional operator settings.

Overall, the presented two-stage latent-space Bayesian paradigm advances physics-informed probabilistic modeling by combining generative neural processes and rigorous physics constraints with efficient, data-driven posterior estimation. This enables robust, uncertainty-aware inference in both traditional and PDE-agnostic physical settings, as demonstrated across regression, inversion, and spatio-temporal prediction tasks (Meng et al., 2021).