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Uncertainty Quantification for Physics-Informed Neural Networks with Extended Fiducial Inference

NeurIPS 2025 · Probabilistic and scientific modeling › Uncertainty and Gaussian processes · approach: Single-model predictive uncertainty · 0 citations (OpenAlex)

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Proposes a narrow-neck hyper-network extension of extended fiducial inference (EFI) that gives honest, prior-free uncertainty quantification for PINNs, validated on a 1-D Poisson equation against Bayesian PINNs and dropout.

Problem

Uncertainty quantification for physics-informed neural networks: Bayesian priors and dropout rates cannot be calibrated without extra information, so confidence sets are not honest.

Approach

Extends EFI to large-scale models by using a narrow-neck w-network (stochastic neck layer, StoNet-style) so that consistency of the inverse mapping holds without sparse priors, letting parameter number exceed sample size; random errors are imputed via adaptive SGLD and parameter uncertainty is propagated through the estimated inverse function.

Key results

• EFI achieves 95.17% coverage with CI-width 0.0504 on 1-D Poisson, averaged over 100 runs, closest to the nominal 95% level.
• Dropout (5%) yields inflated coverage 98.93% with CI-width 0.66; Bayesian with sigma_f=0.0005 yields coverage 58.84%, showing sensitivity to hyperparameters.
• Concrete Dropout attains MSE comparable to EFI (0.000184 vs 0.000148) but coverage only 11.85%.

Measured effects reported in the paper

Surprising findings

• Concrete Dropout matches EFI's MSE yet severely under-covers (11.85%), indicating underestimated predictive uncertainty.

Limitations

• Experiments are limited to relatively small-scale models; scaling to ResNets/CNNs is only proposed via transfer learning, untested.
• Consistency theorem relies on strong regularity and narrow-neck conditions (d_h much smaller than n) and assumptions in the appendix.
• Demonstrated on few PDE examples (1-D Poisson, Montroll, P-FKPP) with a single fixed hyper-parameter regime.

Experimental setup

Datasets: simulated 1-D Poisson data Chinese hamster V79 fibroblast tumor cell growth data scratch assay data

Benchmarks: 1-D Poisson equation

Metrics: MSE coverage rate CI-width

Baselines: PINN (no dropout) MC dropout Concrete Dropout Bayesian PINN (HMC)

Builds on: extended fiducial inference adaptive stochastic gradient MCMC stochastic neural network (StoNet) physics-informed neural networks concrete dropout

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