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arxiv: 2606.03355 · v1 · pith:3Y2LUIG7new · submitted 2026-06-02 · 💻 cs.LG

APIC: Amortized Physics-Informed Calibration using Neural Processes

Aishwarya Venkataramanan , Sai Karthikeya Vemuri , Joachim Denzler This is my paper

Pith reviewed 2026-06-28 11:01 UTC · model grok-4.3

classification 💻 cs.LG
keywords amortized inferencephysics-informed calibrationneural processesKennedy-O'Hagan frameworkmodel discrepancyBayesian calibrationdifferentiable physicsinverse problems
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The pith

APIC uses Neural Processes to amortize Kennedy-O'Hagan calibration across families of related physical systems, separating instance-specific parameters from shared model discrepancies.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents Amortized Physics-Informed Calibration (APIC) as a scalable extension of the Kennedy-O'Hagan framework for physics models that contain systematic errors. It employs Neural Processes with a two-branch latent structure to perform Bayesian inference at the population level rather than fitting each system instance separately. This setup integrates differentiable physics simulations to enable fast calibration of new realizations from limited observations while also recovering uncertainty estimates. Experiments across mechanical, ecological, and PDE examples show gains in parameter accuracy and consistent mapping of the discrepancy patterns that standard per-instance methods miss.

Core claim

APIC employs a two-branch latent architecture within Neural Processes to disentangle instance-specific physical parameters from shared state-dependent structural discrepancies, enabling scalable Bayesian calibration of physics models with misspecifications by integrating differentiable physics into the amortized inference process.

What carries the argument

The two-branch latent architecture of the Neural Process, which separates instance-specific physical parameters from shared state-dependent structural discrepancies while embedding differentiable physics.

If this is right

  • New realizations can be calibrated rapidly without retraining the model on each one.
  • Uncertainty in both parameters and discrepancy estimates is obtained during inference.
  • Parameter recovery improves relative to non-amortized calibration baselines on the tested oscillator, Lotka-Volterra, and advection-diffusion problems.
  • The shared discrepancy component is recovered consistently across realizations even when physics is misspecified.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same amortized structure could be applied to other families of inverse problems where simulations are expensive but many related instances exist.
  • Real-time parameter tracking becomes feasible in settings such as sensor networks or control systems that generate repeated observations of similar dynamics.
  • The approach implicitly treats discrepancy modeling as a meta-learning task over populations of physical systems.

Load-bearing premise

The two-branch latent architecture can reliably disentangle instance-specific physical parameters from shared state-dependent structural discrepancies without additional labels or assumptions on the discrepancy form.

What would settle it

A test case in which the discrepancy structure changes across instances in a manner not capturable by a single shared state-dependent function, causing the method to produce inconsistent parameter estimates or fail to identify the discrepancy.

Figures

Figures reproduced from arXiv: 2606.03355 by Aishwarya Venkataramanan, Joachim Denzler, Sai Karthikeya Vemuri.

Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Qualitative correction pipeline for the damped spring system. From left to right: sparse context observations; [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Qualitative correction pipeline for the Lotka-Volterra ODE System. From left to right: sparse context observations; [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Qualitative correction pipeline for the advection–diffusion system. From left to right: sparse context observations; [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Qualitative reconstruction results for the Lotka–Volterra system. Ground-truth predator and prey populations [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Qualitative discrepancy prediction for the Lotka–Volterra system. The learned discrepancy for predator and prey [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Qualitative reconstruction for the advection–diffusion system using APIC-LNP. Each panel shows the spatial field [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Discrepancy analysis for the advection–diffusion system. The plot shows the predicted discrepancy, the ground [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Effect of context set size on predictive performance for the damped spring system. Reconstruction, discrepancy, [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Effect of observation noise on predictive performance for the damped spring system. Reconstruction, discrepancy, [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Ablation analysis on the influence of λθ (green) and λδ (yellow) on (a) reconstruction MAE, (b) Discrepancy MAE and (c) Physics parameter θ MAE. With increasing values of λθ and λδ, the MAE values decrease [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
read the original abstract

Physics models are inherently imperfect due to misspecified or missing mechanisms, resulting in systematic discrepancies between model predictions and real-world observations. The Kennedy-O'Hagan (KOH) framework addresses this issue through explicit discrepancy modeling. However, its non-amortized, per-instance formulation limits scalability across families of related systems. We introduce Amortized Physics-Informed Calibration (APIC), a population-level extension of KOH that leverages Neural Processes to perform scalable Bayesian inference across realizations. Our framework employs a two-branch latent architecture to disentangle instance-specific physical parameters from shared, state-dependent structural discrepancies. By integrating differentiable physics into an amortized inference backbone, APIC enables rapid calibration of unseen realizations from sparse observations while quantifying uncertainty. Experiments on the damped spring oscillator, the Lotka-Volterra system, and the advection-diffusion PDE with misspecified physics demonstrate improved parameter recovery and consistent identification of the systemic discrepancy structure compared to other calibration approaches.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces Amortized Physics-Informed Calibration (APIC) as a population-level extension of the Kennedy-O'Hagan framework. It uses Neural Processes with a two-branch latent architecture to perform amortized Bayesian inference that disentangles instance-specific physical parameters from shared state-dependent structural discrepancies in misspecified physics models. The method integrates differentiable physics into the inference backbone to enable rapid calibration of unseen realizations from sparse observations with uncertainty quantification. Experiments on the damped spring oscillator, Lotka-Volterra system, and advection-diffusion PDE are reported to show improved parameter recovery and consistent discrepancy identification relative to other calibration approaches.

Significance. If the two-branch architecture reliably achieves the claimed disentanglement, the work would offer a scalable alternative to per-instance KOH calibration, with potential impact on scientific machine learning tasks involving families of related physical systems. The amortized formulation and uncertainty quantification are strengths, but the significance hinges on whether the separation of latents is demonstrated beyond the reported experiments.

major comments (2)
  1. [Method (two-branch latent architecture)] The central claim that the two-branch latent architecture disentangles instance-specific physical parameters from shared discrepancies (abstract and method description) lacks an identifiability analysis, auxiliary loss, or theoretical guarantee. Standard NP encoders produce entangled representations; without explicit inductive bias or supervision, the branches can trade off information under sparse observations and misspecified physics, which directly undermines the reported parameter recovery results.
  2. [Experiments] The experiments claim improved parameter recovery and consistent discrepancy identification, but without details on how separation is verified (e.g., latent traversal, ablation on branch usage, or comparison to entangled baselines), it is unclear whether the performance gains arise from true disentanglement or from the amortized fitting process itself.
minor comments (2)
  1. [Method] Notation for the two latent branches and their conditioning on observations vs. states should be clarified with explicit equations to avoid ambiguity in the amortized inference procedure.
  2. [Method] The paper should include a clear statement of the precise form of the discrepancy function and how it is parameterized within the NP decoder.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address the major comments point-by-point below and propose revisions to strengthen the claims regarding disentanglement.

read point-by-point responses

  1. Referee: [Method (two-branch latent architecture)] The central claim that the two-branch latent architecture disentangles instance-specific physical parameters from shared discrepancies (abstract and method description) lacks an identifiability analysis, auxiliary loss, or theoretical guarantee. Standard NP encoders produce entangled representations; without explicit inductive bias or supervision, the branches can trade off information under sparse observations and misspecified physics, which directly undermines the reported parameter recovery results.

    Authors: We agree that a formal identifiability analysis is absent from the current manuscript. The two-branch design is intended to provide an inductive bias by routing instance-specific information through one latent branch and state-dependent discrepancy through the other, with the physics model applied only to the parameter branch. However, without additional supervision or loss terms, complete separation is not guaranteed. In the revision, we will add an explicit comparison to an entangled single-branch baseline to quantify the benefit of the architecture. We will also discuss the limitations of relying on architectural separation without theoretical guarantees. revision: partial

  2. Referee: [Experiments] The experiments claim improved parameter recovery and consistent discrepancy identification, but without details on how separation is verified (e.g., latent traversal, ablation on branch usage, or comparison to entangled baselines), it is unclear whether the performance gains arise from true disentanglement or from the amortized fitting process itself.

    Authors: We acknowledge the need for more rigorous verification of the disentanglement. The current experiments demonstrate improved parameter recovery and discrepancy identification compared to baselines, but do not include ablations on the branch architecture. We will revise the experimental section to include: (1) an ablation study using a single latent branch (entangled) model, (2) qualitative analysis of the latent representations (e.g., how the discrepancy latent varies with state), and (3) sensitivity analysis under sparse observations. This will help clarify whether the gains stem from the disentanglement. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces APIC as an amortized extension of the Kennedy-O'Hagan framework via Neural Processes and a two-branch latent architecture for disentangling parameters from discrepancies. The abstract and description present this as a proposed architecture whose performance is evaluated on experiments with damped spring, Lotka-Volterra, and advection-diffusion systems. No quoted equations, self-citations, or fitted inputs reduce the central claims (rapid calibration, uncertainty quantification, or disentanglement) to tautological re-expressions of the inputs by construction. The architecture is offered as an inductive choice rather than derived from prior results that presuppose the target outcome. This is the common case of a self-contained methodological contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard domain assumptions in physics-informed machine learning plus the effectiveness of the newly introduced two-branch architecture; no free parameters or invented physical entities are named in the abstract.

axioms (2)
  • domain assumption Physics models are inherently imperfect due to misspecified or missing mechanisms
    Opening sentence of the abstract.
  • domain assumption Neural Processes can perform scalable Bayesian inference across a population of related systems
    Core modeling choice stated in the abstract.
invented entities (1)
  • two-branch latent architecture no independent evidence
    purpose: Disentangle instance-specific physical parameters from shared state-dependent structural discrepancies
    Introduced as the key architectural component of APIC.

pith-pipeline@v0.9.1-grok · 5697 in / 1381 out tokens · 40565 ms · 2026-06-28T11:01:45.334901+00:00 · methodology

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Reference graph

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