Evidential PINNs for Uncertainty-Aware PDE Modeling

• Evidential PINNs are neural network models that integrate physical laws with evidential deep learning to quantify both epistemic and aleatoric uncertainty in solving PDEs.
• They employ a composite loss function combining data fidelity and physics residuals—augmented by evidential priors—to ensure robust parameter inference and model calibration.
• Empirical results highlight E-PINNs' superior coverage probabilities and low calibration errors, making them effective for forward simulation, inverse problem solving, and scientific data mining.

Evidential Physics-Informed Neural Networks (E-PINN) represent a convergent methodology combining uncertainty quantification with rigorous physical constraints in neural network-based PDE modeling. This class of models extends the standard physics-informed neural network framework by integrating evidential deep learning principles, yielding predictive models that estimate both the solution to physical systems and detailed measures of epistemic and aleatoric uncertainty. E-PINN approaches address model calibration, robustness to noise, parameter inference, and the reliability of predictions, with applicability to forward simulation, inverse problem solving, and scientific data mining. Current implementations demonstrate superior empirical coverage probabilities, calibrated uncertainty intervals, and real-world applicability in fields like medical modeling.

1. Mathematical Foundations and Theory

E-PINN builds on the core architecture of PINNs, in which a neural network $\mathcal{M}(w)$ is trained to reconstruct system outputs $y$ across the domain $x$ by minimizing a composite loss integrating both data fidelity and the physics residual. The evidential component emerges from imposing a higher-order probabilistic model on the network outputs and unknown PDE parameters.

For scalar response $u$, E-PINN models its distribution with a normal-inverse-gamma (NIG) hierarchy parameterized by $\{\gamma, \alpha, \beta, \nu\}$:

• Predictive mean: $\mathbb{E}[u] = \gamma$
• Predictive variance: $$\mathrm{Var}[u] = \frac{\beta}{\alpha-1}\left(1 + \frac{1}{\nu}\right)$$

Model training involves minimizing the negative log marginal likelihood:

$$-\log P(\mathcal{D} | \mathcal{M}(w); \gamma, \alpha, \beta, \nu)$$

E-PINN further imposes priors on unknown physical parameters $\Omega$ of the PDE and the variance weight $\sigma_R^2$ of the residual term. The joint loss function thereby enforces:

$$L = -\log[P(\mathcal{D} | \mathcal{M}(w)) \cdot P(\mathcal{M}(w) | \Omega) \cdot \pi(\Omega) \cdot \pi(\sigma_R^2)]$$

where $P(\mathcal{M}(w)|\Omega)$ is defined from the residual $\mathcal{R}(\partial_y, y, x, \Omega)$ of the discretized PDE.

2. Evidential Deep Learning Integration

E-PINN introduces evidential priors, learning the parameters $\{\gamma, \alpha, \beta, \nu\}$ directly from data, enabling closed-form uncertainty propagation. The outputs encode both mean predictions and associated uncertainties, distinguishing them from deterministic PINNs and Bayesian PINNs that often require Monte Carlo sampling or variational approximations.

When applied to inverse problems, E-PINN replaces fixed PDE coefficients (e.g., $\kappa$) with evidential random variables, propagating uncertainty in both the solution and estimated parameters. Marginalization over these evidential priors yields residuals and parameter posteriors:

$$\mathcal{G}(u,\dots,\mu_\kappa) = \mathcal{L}_1^2 + \mathcal{L}_2^2 (\mathrm{Var}(\kappa) + \overline{\kappa}^2) + 2\overline{\kappa}\mathcal{L}_1\mathcal{L}_2$$

These mechanisms render E-PINN attractive for parameter inference in ill-posed or noisy settings.

3. Loss Function Construction and Regularization

The loss function in E-PINN consists of:

• Marginal evidential loss from the NIG marginal likelihood,
• Physics residual loss evaluated using numerical discretizations (e.g., finite differences, POD-Galerkin, DEIM for nonlinear terms) at collocation points,
• Information-theoretic regularizers such as Kullback-Leibler divergence between learned inverse-gamma posteriors and weakly informative reference priors.

The information-theoretic regularizer is mathematically expressed as:

$$\mathcal{L}_{kl} = |u_i - \gamma|(2\nu + \alpha) \{ \log[(\beta_r^\alpha \Gamma(\alpha))/\beta^\alpha] + \gamma_e (\alpha-1) + (\beta-\beta_r)/\beta_r \}$$

with $\beta_r$ setting the reference scale, crucial for calibration under data noise and out-of-distribution generalization.

The weighting of the PDE residual term $\sigma_R^2$ is treated as a learnable parameter regularized via inverse-gamma priors, ensuring its adaptive evolution during training.

4. Uncertainty Quantification and Parameter Inference

E-PINN achieves uncertainty quantification through analytic propagation of the NIG parameters and learned priors on unknown coefficients. Closed-form confidence intervals can be computed for field variables and PDE parameters.

Parameter inference proceeds by maximizing the posterior:

$$f_p(\Omega|\mathcal{D},\mathcal{M}(w)) \propto P(\mathcal{M}(w) | \Omega)\,\pi(\Omega)$$

Upon training, E-PINN produces interval estimates for physical coefficients (e.g., source locations, diffusivities, kinetic rates), often with empirical coverage probabilities closely matching nominal expectations.

Performance in standard benchmarks, such as Poisson, nonlinear reaction-diffusion, and Fisher-KPP equations, demonstrates low calibration error and robustness to synthetic and in situ data noise.

5. Implementation Algorithms and Architectural Variants

E-PINN models leverage feedforward or recurrent neural network backbones, optionally combining discretized residual penalties that allow leaner inputs (e.g., only aerodynamics coefficients in reduced-physics models, per (Halder et al., 2023)). The use of time-stepping schemes and projection methods (e.g., backward Euler, POD-Galerkin, DEIM) enhances flexibility for time-dependent and high-dimensional systems.

Recent architectures integrate meta-network layers and adaptive nonlinear transforms to capture shock interactions and discontinuities, evidencing physical constraints (e.g., entropy, Rankine–Hugoniot) within the loss structure (Ji et al., 3 Jan 2025).

For uncertainty quantification, auxiliary networks (epinets) can be attached to pre-trained PINNs, with the final prediction given by:

$u_\theta(x,z) = u_\xi(x) + e_\eta(\tilde{x},z)$

with $z$ sampled from an epistemic reference distribution. This augments the main prediction with epistemic and prior contributions and enables accurate empirical coverage in noisy regression and PDE tasks (Nair et al., 25 Mar 2025).

The training protocol typically proceeds in two phases: data-only evidential fit, then full data-plus-PDE training with adaptive weighting. Priors for unknown PDE parameters and residual strengths are constructed based on data-inferred model discrepancy and regularized by information-theoretic metrics.

6. Benchmark Evaluations and Practical Applications

In controlled benchmarks, E-PINN matches or exceeds calibration of Bayesian PINN and Deep Ensemble methods. For the 1D Poisson equation with a Gaussian source, E-PINN correctly infers source location and spread with narrow credible intervals (e.g., $x_0 = 0.334^{+0.013}_{-0.008}$). In the Fisher–KPP reaction-diffusion system, parameter estimates tracked ground truth ($r = 1.60^{+0.05}_{-0.03}$, $D = 6.26^{+1.22}_{-1.30}$). Calibration errors are consistently low, and empirical coverage probabilities remain close to nominal, with high p-values in MC goodness-of-fit tests.

Real-world applicability is demonstrated on clinical glucose-insulin data via the Bergman model. E-PINN infers distributions for physiological indices (e.g., insulin sensitivity $S_I = p_3/p_2$) capable of discriminating healthy vs diabetic states by several orders of magnitude. The uncertainty measures provide meaningful bounds for medical decision making and scientific interpretation.

7. Comparative Context and Future Directions

Relative to Bayesian PINNs—which require expensive posterior sampling or Laplace approximations—E-PINNs present a closed-form, efficient alternative with adaptive calibration. Compared to Dropout-PINNs, E-PINNs yield more consistent uncertainty bands and empirical coverage less sensitive to hyperparameter selection. E-PINNs maintain boundary adherence and robustness in the presence of out-of-distribution test scenarios.

Emerging directions involve extending E-PINN frameworks across broader PDE collections (e.g., PINNacle benchmarks (Tan et al., 27 Jan 2025)), integrating more advanced physical priors (e.g., conservation laws, multi-physics), and validating the theory for operator learning and high-dimensional scientific discovery.

A plausible implication is that E-PINN methodology will form the backbone for next-generation scientific machine learning systems, capable of interpretable, trustworthy modeling in domains where physical law and uncertainty coexist as central tenets.

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Summary Table: E-PINN Components and Roles

Component	Role	Notation
Evidential output	Predictive mean/variance quantification	$\{\gamma, \alpha, \beta, \nu\}$
Physics residual loss	Enforces PDE constraints	$\mathcal{R}(\partial_y, y, x, \Omega)$
PDE parameter prior	Uncertainty in unknown PDE coefficients	$\pi(\Omega)$
Residual weight prior	Adaptive weighting, regularized via inverse-gamma	$\pi(\sigma_R^2; \alpha_r, \beta_r)$
Information regularizer	Calibration and robustness	KL divergence

Evidential Physics-Informed Neural Networks thus formalize the interplay between data evidence, physical constraint, and uncertainty, establishing a framework for certified and interpretable machine learning in scientific and engineering contexts.