PINNfluence: Interpreting PINNs Through Influence Functions

Conference: ICML 2026
arXiv: 2409.08958
Code: https://github.com/aleks-krasowski/pinnfluence
Area: Scientific Computing / Physics-Informed Neural Networks / Interpretability
Keywords: PINN Diagnostics, Influence Functions, Training Data Attribution, Loss Decomposition, Temporal Causality Metrics

TL;DR

This paper extends Influence Functions, a training data attribution method, to Physics-Informed Neural Networks (PINNs), proposing PINNfluence. By using linearized leave-one-out perturbation estimation, it attributes the prediction, loss, or physical quantities of a PINN simultaneously to each training point and each loss component. Based on this, it constructs a set of diagnostic metrics (loss component ratios, cancellation scores, temporal causality metrics, etc.) that consistently distinguish between "well-trained" and "poorly-trained" PINNs across five time-dependent PDEs, providing structural diagnostics that residual analysis fails to identify.

Background & Motivation

Background: PINNs (Raissi 2019) embed PDE residuals as soft constraints into the NN training objective, using a unified loss \(\mathcal{L}(\theta)=\lambda_\text{pde}\!\sum L_\text{pde}+\sum_k\lambda_{bc,k}\!\sum L_{bc,k}\) to learn a function \(\phi(x;\theta)\) approximating the solution of an IBVP. While widely applied in fluids, electromagnetics, epidemiology, and optics, training failures (propagation failure, overly strong initial conditions, ill-conditioned loss landscapes) are extremely common, and the community has historically relied on "phenomenological diagnostics."

Limitations of Prior Work: (1) Almost all understanding of PINNs is based on training dynamics (gradient flow analysis, NTK, loss reweighting) rather than post-hoc interpretability—identifying that a model has issues, but not why a specific prediction originates from certain training data or which loss term dominates it. (2) Traditional verification fails for PINNs: low training loss does not equal a correct PDE solution—PINNs often converge to trivial solutions with low residuals but incorrect physics (Daw 2023, Rohrhofer 2023), which residual checks cannot detect. (3) While XAI is mature in vision/NLP (LRP, IF, SAE), no dedicated post-hoc interpretation methods exist for PINNs.

Key Challenge: PINN failures are often structural (over-reliance on initial conditions, failed boundary propagation, information disconnection at temporal cross-sections), but scalar metrics like loss or residuals flatten this structure. To see the structure, one must unfold three axes: "which training point, which loss term, and which spatio-temporal region."

Goal: (1) Generalize the classical Influence Functions of Koh & Liang (2017) to the composite loss and arbitrary differentiable objectives of PINNs; (2) Provide influence-based diagnostic metrics capable of structurally distinguishing "well-trained" from "poorly-trained" models; (3) Verify the stability of these metrics across multiple PDEs and optimizers.

Key Insight: The authors note that PINN loss is naturally a weighted sum of components, and IF is linear with respect to loss parameters. This means a single Hessian-vector product calculation allows influence to be "automatically" decomposed into each \(L_i\). By combining "training point \(\to\) region" and "region \(\to\) training point" aggregation directions, one can construct attribution maps at point-point, point-region, and region-region granularities.

Core Idea: Extend IF from "loss-to-loss" attribution to "arbitrary differentiable composite loss \(L\) to arbitrary differentiable output \(f\)" attribution \(\operatorname{Inf}_{\theta_0}^{L\to f}(x,z)=-\nabla_\theta f(z;\theta_0)^\top \mathcal{H}_{\theta_0}^{-1}\nabla_\theta L(x;\theta_0)\). Utilize linearity and additivity to decompose the results into every loss component and spatio-temporal region of the PINN.

Method

PINNfluence does not alter the PINN training process; it is a post-hoc analysis framework. Given a pre-trained PINN \(\phi(\cdot;\theta_0)\) and its training set \(\mathcal{X}=\mathcal{X}_\text{pde}\cup\bigcup_k\mathcal{X}_{bc,k}\), it computes \(\operatorname{Inf}\) to aggregate diagnostic metrics.

Overall Architecture

• Input: Trained PINN \(\phi\), training set \(\mathcal{X}\) (including PDE collocation and IC/BC points), target quantity of interest \(f\) (can be prediction \(\hat{u}\), a loss component \(L_i\), or physical observables), and test points/regions.
• Mechanism: Pair the IHVP \(\mathcal{H}_{\theta_0}^{-1}\nabla_\theta L(x;\theta_0)\) with \(\nabla_\theta f(z;\theta_0)\) using low-rank Arnoldi approximation and Hessian-vector products to avoid explicit Hessian construction.
• Three-layer Granularity:
  • Point-to-Point: \(\operatorname{Inf}_{\theta_0}^{L\to f}(x,z)\);
  • Point-to-Region / Region-to-Point: Summing over \(z\) or \(x\) in a region;
  • Region-to-Region: Double summation with normalization for proportional metrics.
• Output: (1) Point-to-point influence heatmaps; (2) Loss component decomposition ratios \(r_{L_i}\) and cancellation scores \(\kappa\); (3) Spatio-temporal metrics, such as the temporal causality metric \(\eta\).

The pipeline structure computes the influence once and branches into multiple diagnostics. The core calculation is the generalized influence function \(\operatorname{Inf}^{L\to f}\), which aggregates across three granularities before splitting into "Loss Component" and "Spatio-temporal" diagnostic families to form a structural judgment of "well-trained vs. poorly-trained."

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["Input: Trained PINN φ + Training Set X<br/>(PDE collocation / IC / BC points) + Target f"]
    A --> B["Generalized Influence Function (Any diff. L → Any diff. f)<br/>Inf = −∇f · H⁻¹ · ∇L (Arnoldi Low-rank IHVP + HVP)"]
    B --> C["3-Layer Aggregation: Point-Point / Point-Region / Region-Region<br/>(Bilinear in loss, additive)"]
    C --> D["Loss Component Ratios + Cancellation Score κ"]
    C --> E["Temporal Causality η + Regional Normalized Metric ρ"]
    D --> F["Structural Diagnosis: Well-trained vs. Poorly-trained"]
    E --> F

Key Designs

1. Extension from "Loss-to-Loss" to "Arbitrary Target + Composite Loss"

Classical Influence Functions (Koh & Liang 2017) only answer how adding/removing a point affects the training loss, which is insufficient for PINNs. PINN failure diagnosis requires knowing "which loss term and which region dominates a specific prediction." The authors generalize attribution to:

\[\operatorname{Inf}_{\theta_0}^{L\to f}(x,z)=-\nabla_\theta f(z;\theta_0)^\top\mathcal{H}_{\theta_0}^{-1}\nabla_\theta L(x;\theta_0),\]

where \(f\) can be \(\hat u\), a specific component loss, or any physical observable. They prove this first-order approximates the leave-one-out re-training effect (Thm 2.2). A critical relaxation replaces "strict local minimum + strong convexity" with "non-degenerate stationary point + invertible Hessian," as NN training often settles at saddle points. This allows IF to decompose PINN composite losses \(\mathcal{L}=\lambda_\text{pde}L_\text{pde}+\sum_k\lambda_{bc,k}L_{bc,k}\) and explain diagnostic errors.

2. Loss Component Decomposition + Cancellation Score \(\kappa\)

IF is linear with respect to \(L\) parameters (Corollary 2.3): \(\operatorname{Inf}^{\sum_i\alpha_iL_i\to f}=\sum_i\alpha_i\operatorname{Inf}^{L_i\to f}\). Thus, total influence is decomposed into components for "free." Relative contribution is defined as \(r_{L_i}(x,z)=\frac{|\operatorname{Inf}^{L_i\to f}|}{\sum_j|\operatorname{Inf}^{L_j\to f}|}\), and the cancellation score as:

\[\kappa(x,z)=1-\frac{\big|\sum_j\operatorname{Inf}^{L_j\to f}\big|}{\sum_j|\operatorname{Inf}^{L_j\to f}|}.\]

\(\kappa\) determines if decomposition is reliable: if components cancel out (large \(\kappa\)), \(r_{L_i}\) might be misleading; if no cancellation occurs, \(r_{L_i}\) represents a clear contribution ratio.

3. Temporal Causality Metric \(\eta\) + Regional Normalized Metric \(\rho\)

The temporal relationship between training point \(x_t\) and test point \(z_t\) is encoded to quantify if the PINN "looks from the past to the future":

\[\eta_{\theta_0}^{L\to f}(R_\text{tr},R_\text{te})=1-\frac{1}{|R_\text{te}|}\sum_{z\in R_\text{te}}\frac{\sum_{x\in R_\text{tr}:x_t\le z_t}|\operatorname{Inf}^{L\to f}(x,z)|}{\sum_{x\in\mathcal{X}_\text{train}}|\operatorname{Inf}^{L\to f}(x,z)|},\]

A small \(\eta\) implies influence mostly comes from the past (causally aligned), whereas a large \(\eta\) suggests future influence. While intuition suggests well-trained PINNs should follow PDE causality, the empirical finding is the opposite: well-trained models show \(\eta\approx\bar t\) (near uniform), while failed models show strong "past dominance" (abnormal IC persistence).

Loss & Training

No new training loss is introduced—PINNfluence performs post-hoc analysis. Implementation relies on PyHessian-style HVP + Schioppa 2022's Arnoldi low-rank approximation to estimate \(\mathcal{H}^{-1}\), avoiding \(O(p^2)\) explicit Hessian construction. Proximal Bregman response functions (PBRF, Bae 2022) are used for reliability checks. It is verified across optimizers like NNCG (Rathore 2024) and SOAP (Vyas 2025).

Key Experimental Results

Settings: 5 time-dependent PDEs (Heat, Allen-Cahn, Burgers', Wave, Drift-Diffusion) and 2 steady-state PDEs (Poisson, Navier-Stokes). Each problem includes "well-trained" vs. "poorly-trained" configurations across 10 seeds.

Main Results

Problem	\(\bar{t}\) (Baseline)	Well-trained \(\eta\) (pred)	Poorly-trained \(\eta\) (pred)	Diagnostic Conclusion
Heat	0.46	0.33 ± 0.02	0.26 ± 0.06	Failed models bias toward past (IC dominant)
Allen-Cahn	0.43	0.50 ± 0.02	0.32 ± 0.05	Failed models show significant \(\eta\) decrease
Burgers'	0.43	0.41 ± 0.02	0.28 ± 0.02	Same as above
Drift-Diffusion	0.46	0.46 ± 0.04	0.21 ± 0.06	IC influence overwhelms, corresponding to propagation failure
Wave	0.43	0.41 ± 0.03	0.11 ± 0.02	Largest gap; failed models almost entirely anchored to IC

→ Across all 5 problems, \(\eta\) for poorly-trained models is significantly lower than for well-trained models, with small standard deviations across seeds.

Ablation Study

Dimension	Experimental Setup	Key Finding
Loss Component \(\bar{r}_{L_\text{ic}}\)	5 PDEs × 50 time bins	Well-trained: IC share starts at \(\approx 0.25\) and decays; Poorly-trained: IC share stays high and plateaus after \(t\approx 0.4\).
Training Point Count Scan	5 problems	PINNfluence metrics transition smoothly; the curve shape is problem-dependent, revealing data demand structures.
Optimizer Independence	Adam / NNCG / SOAP	Distinguishing patterns for \(\eta\) and \(\bar{r}_{L_i}\) remain consistent across types of optimizers.
Hessian Reliability	Low-rank Arnoldi vs PBRF vs Gradient-only	Arnoldi faithfully recovers the projected inverse Hessian even under ill-conditioning, significantly outperforming gradient-only baselines.
Cancellation Score \(\kappa\)	Loss term decomposition	\(\kappa\) is small at most points, making \(r_{L_i}\) robust; high \(\kappa\) points indicate "competing constraints."

Key Findings

• Structural signals outweigh residual signals: Residuals only show errors; PINNfluence localizes "why," e.g., IC influence over-persisting or BC being ignored.
• Counter-intuitive causality: Well-trained PINNs do not strictly follow physical causality (\(\eta\) near uniform). Strong causal alignment is often a byproduct of IC overfitting.
• Consistency across tasks: IC influence decay patterns consistently distinguish training quality across distinct PDEs, suggesting potential as a universal diagnostic.

Highlights & Insights

• The generalization of IF is the pivotal step: Converting "data \(\to\) physical quantity" attribution into a single IHVP turns IF from a classifier debugger into a scientific computing tool.
• \(\kappa\) as a "decomposition self-check": This addresses the risk of misleading linear sums due to signed cancellations, a detail transferable to LRP/Shapley methods.
• Well-trained PINNs are globally optimized: They approximate the PDE solution as a global variational problem rather than a time-stepper, explaining why they don't exhibit strict physical causality in sensitivity patterns.

Limitations & Future Work

• First-order approximation: Subject to \(O(1/N^2)\) remainders; large perturbations or extreme ill-conditioning may introduce bias.
• PINN Hessian ill-conditioning: While low-rank Arnoldi helps, singular directions may still amplify errors in complex PDEs.
• Computational overhead: Linear in \(|\mathcal{X}_\text{train}|\times|R_\text{te}|\); potentially expensive for massive 3D time-varying datasets.
• Loss of directionality: Aggregating absolute values \(|\operatorname{Inf}|\) prevents noise but loses "promotion vs. inhibition" information.

Related Work & Insights

• vs. Koh & Liang (2017): Extends IF beyond loss-to-loss attribution and strong convexity assumptions to composite losses.
• vs. PINN Training Dynamics (Wang 2022, Rathore 2024): Complementary; these study "why it fails during training," while PINNfluence studies "how it fails structurally" post-training.
• vs. Neural Operator Interpretability (MacMillan 2025): Approaches from training data attribution rather than mechanistic latent feature analysis.

Rating

• Novelty: ⭐⭐⭐⭐⭐ First attribution framework for PINNs; generalized IF has broad utility.
• Experimental Thoroughness: ⭐⭐⭐⭐ Solid across 5+ PDEs and multiple optimizers; lacks 3D benchmarks.
• Writing Quality: ⭐⭐⭐⭐⭐ Mature progression from theory to metrics to experiments.
• Value: ⭐⭐⭐⭐⭐ Fills a gap in sci-ML interpretability with potential for diagnostic-driven training.

Related Papers

• [ICML 2026] Generative Neural Operators Through Diffusion Last Layer
• [NeurIPS 2025] Neural Green's Functions
• [NeurIPS 2025] Scaling Laws and Pathologies of Single-Layer PINNs: Network Width and PDE Nonlinearity
• [NeurIPS 2025] One-Shot Transfer Learning for Nonlinear PDEs with Perturbative PINNs
• [NeurIPS 2025] Towards Universal Neural Operators through Multiphysics Pretraining