APIC: Amortized Physics-Informed Calibration using Neural Processes

Technical Summary: Amortized Physics-Informed Calibration using Neural Processes (APIC)

Problem Statement

Physical simulators are essential for scientific modeling but inherently suffer from systematic discrepancies with real-world observations due to misspecified mechanisms, missing physics, or numerical approximations. The Kennedy–O'Hagan (KOH) framework addresses this by decomposing observations into a physical simulator output, a systematic discrepancy term, and observational noise. However, classical KOH relies on non-amortized, per-instance Bayesian inference (typically using Gaussian Processes and MCMC), which is computationally expensive and does not scale to families of related systems. Furthermore, without strong structural assumptions, the discrepancy term can confound the inference of physical parameters, leading to non-identifiability where model errors mimic parameter misspecification.

While hybrid approaches like Physics-Informed Neural Networks (PINNs) and Neural Processes (NPs) have emerged, they often lack a unified probabilistic formulation that simultaneously amortizes inference across a population and explicitly disentangles physical parameters from structured model discrepancies.

Methodology: APIC Framework

The authors propose Amortized Physics-Informed Calibration (APIC), a population-level extension of the KOH framework that leverages Neural Processes (NPs) to perform scalable Bayesian inference.

1. Problem Formulation
APIC generalizes the KOH equation to a family of related realizations $Y = \{y_i\}$. For each realization $i$, observations are modeled as:
$$y_i = F(x_i; \theta_i) \circledast \delta(x_i; \theta_i) + \epsilon_i$$
where $F$ is a differentiable physics solver, $\theta_i$ are instance-specific physical parameters, and $\delta$ is a state-dependent discrepancy function. The operator $\circledast$ allows for both additive and multiplicative corrections.

2. Dual-Latent Architecture
The core of APIC is a dual-branch latent encoder architecture designed to disentangle instance-specific physics from shared structural discrepancies:

• Physics Latent Branch: Encodes sparse context observations into a latent variable $z^{(\theta)}_i$, which is mapped to instance-specific solver parameters $\hat{\theta}_i$. This branch drives the differentiable simulator $F$.
• Discrepancy Latent Branch: Encodes the same context into a latent variable $z^{(\delta)}_i$, which parameterizes a state-dependent discrepancy function $\delta_i$. This branch learns a shared, structural correction model that accounts for unmodeled physics.
• Generative Model: The decoder combines the simulator output $F(x; \hat{\theta}_i)$ and the discrepancy correction $\delta(x; z^{(\theta)}_i, z^{(\delta)}_i)$ to produce predictive distributions with uncertainty estimates.

3. Two-Stage Curriculum Training
To ensure meaningful disentanglement and prevent the discrepancy network from absorbing physical parameter variations, APIC employs a two-stage training strategy:

• Stage 1 (Physics Pretraining): The model is trained on a nominal dataset (generated by the physics solver with known ground-truth parameters) using only the physics branch. The objective is to minimize the supervised loss between predicted and true parameters ($\|\theta_i - \hat{\theta}_i\|^2$), anchoring the physics encoder to physically meaningful structures.
• Stage 2 (Joint Variational Training): The full model (including the discrepancy network) is trained via variational inference on real observation data. The objective maximizes the Evidence Lower Bound (ELBO) while retaining the parameter supervision from Stage 1. A regularization term $L_\delta$ (specifically $L_2$ regularization in experiments) is applied to the discrepancy function to discourage it from capturing variations explainable by the physics encoder.

Key Contributions

1. Unified Framework: Introduction of APIC, which integrates the probabilistic rigor of the KOH framework with the amortized efficiency of deep generative models (Neural Processes) for population-level calibration.
2. Disentangled Architecture: Proposal of a dual-latent architecture that explicitly separates instance-specific physical parameters from structured model discrepancies, enabling physically interpretable inference even under sparse observations.
3. Scalable Inference: Demonstration that APIC achieves rapid calibration for unseen realizations without the need for per-instance optimization, while providing principled uncertainty quantification.

Experimental Results

The authors evaluated APIC on three dynamical systems: a damped spring oscillator, a Lotka–Volterra ODE (predator-prey), and an advection–diffusion PDE. In all cases, the "true" system included structured, state-dependent discrepancies not present in the nominal simulator.

• Performance: APIC variants (CNP, LNP, ANP) consistently outperformed classical KOH-GP, Bayesian Calibration with Physics-Informed Priors (BCPI), and Correction-PINN baselines across reconstruction accuracy (MAE), predictive log-likelihood (LL), and parameter recovery.
• Discrepancy Recovery: APIC successfully identified and compensated for systematic model biases, as visualized in qualitative correction pipelines where the learned discrepancy accurately bridged the gap between the misspecified simulator and ground truth.
• Parameter Identifiability: By disentangling the discrepancy, APIC achieved significantly lower Mean Absolute Error in recovering ground-truth physical parameters compared to baselines, which often suffered from non-identifiability.
• Efficiency: APIC demonstrated orders-of-magnitude improvement in inference speed. For the damped spring system, APIC-LNP required 0.42 ms per realization, compared to 12.6 s for BCPI, 33.6 s for KOH-GP, and 150 s for Correction-PINN (with ensembles).
• Uncertainty Calibration: Stochastic variants of APIC (LNP, ANP) showed improved Expected Calibration Error (ECE), indicating well-calibrated predictive uncertainty.

Significance and Claims

The paper claims that APIC addresses the critical gap between the theoretical rigor of classical calibration and the scalability of deep learning. By unifying amortized inference with physics-informed constraints, APIC enables the rapid calibration of imperfect physical systems across populations of realizations. The authors assert that their approach not only improves predictive accuracy but, crucially, preserves the physical interpretability of the inferred parameters by preventing the "leakage" of physical parameter uncertainty into the discrepancy term. This makes APIC particularly suitable for scientific domains requiring rapid, reliable, and interpretable calibration of complex simulators under sparse data conditions.