🤖 machine learning

naPINN: Noise-Adaptive Physics-Informed Neural Networks for Recovering Physics from Corrupted Measurement

The paper proposes naPINN, a Noise-Adaptive Physics-Informed Neural Network that robustly recovers physical solutions from corrupted measurements by embedding an energy-based model to learn residual distributions and employing a trainable reliability gate to filter outliers without prior knowledge of the noise distribution.

Original authors: Hankyeol Kim, Pilsung Kang

Published 2026-06-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hankyeol Kim, Pilsung Kang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: Learning from a Broken Radio

Imagine you are trying to learn the rules of a complex game (like how a fluid flows or how heat spreads) just by listening to a radio broadcast of the game. This is what scientists call an inverse problem: trying to figure out the hidden laws of physics based on what we observe.

Usually, we use a smart computer program called a PINN (Physics-Informed Neural Network) to do this. Think of a PINN as a very eager student who tries to memorize the game rules by listening to the radio.

The Catch: In the real world, the radio is often broken.

Sometimes the signal is just a little fuzzy (noise).
Sometimes the radio suddenly screams static or plays a completely different song (gross outliers).

If you try to teach a student using a radio that screams static, the student gets confused. They might think the static is part of the game, and they will learn the wrong rules. Standard PINNs are like students who take every sound they hear seriously, even the screaming static. When the data is "corrupted" (full of bad noise), these students fail miserably.

The Solution: The "Noise-Adaptive" Student (naPINN)

The authors of this paper created a new, smarter student called naPINN (Noise-Adaptive Physics-Informed Neural Network).

Instead of blindly trusting every sound, naPINN has a special filter (a "reliability gate"). Here is how it works, step-by-step:

1. The Warm-Up Phase

First, naPINN listens to the radio for a short while without any special filters. It tries to get a general idea of the game. This is like the student sitting in the classroom for the first few minutes to get a feel for the room.

2. The "Sniffer" (The Estimator)

Once the student has a basic idea, it starts paying attention to the mistakes it makes.

If the student predicts a score of 10, but the radio says 10, that's a good match.
If the student predicts 10, but the radio screams "1,000,000!", that's a huge mismatch.

naPINN uses a special tool (called an Energy-Based Model) to analyze these mismatches. It asks: "Is this huge mistake just a weird fluke of the noise, or is this total garbage?"

Normal noise: The tool says, "This is weird, but it fits the pattern of a broken radio."
Gross outliers: The tool says, "This is impossible. This is a broken speaker screaming nonsense."

3. The "Volume Knob" (The Reliability Gate)

This is the magic part. naPINN has a volume knob for every single piece of data it hears.

If the "Sniffer" says a piece of data is reliable, the volume knob is turned up. The student listens closely.
If the "Sniffer" says a piece of data is garbage (an outlier), the volume knob is turned down to almost zero. The student ignores it.

4. The Safety Net (Rejection Cost)

You might think, "Why not just turn the volume down on everything to avoid mistakes?"
The paper adds a rule: You can't ignore everything.
If the student tries to ignore all the data to make the math easy, it gets punished. This forces the student to keep listening to the good parts of the radio while ignoring the bad parts.

What Happened in the Experiments?

The researchers tested this new student on three different "games" (mathematical equations describing physics):

Burgers' Equation (like traffic flow or shockwaves).
Allen-Cahn Equation (like how materials separate, like oil and water).
Reaction-Diffusion System (like how chemicals spread and react).

They deliberately broke the data by adding:

Weird noise: Not just random static, but complex, multi-shaped noise.
Gross outliers: 5%, 10%, or even 15% of the data was replaced with completely wrong numbers.

The Results:

Old Students (Standard PINNs): When the data was corrupted, they failed. They tried to fit the screaming static, and their predictions were wild and wrong.
Other "Robust" Students: Some tried to use different math tricks to handle noise, but they still struggled when the noise was very strange.
naPINN: It was the clear winner. Even when 15% of the data was garbage, naPINN successfully ignored the garbage and learned the correct rules. It reconstructed the physics almost as well as if the data had been clean.

The Bottom Line

The paper claims that naPINN is a framework that can learn physics from messy, broken data without needing to know exactly how the data is broken beforehand.

It does this by:

Learning what "normal" mistakes look like.
Identifying "impossible" mistakes (outliers).
Silencing the impossible mistakes while keeping the good data.

It's like teaching a student to ignore a broken radio's screaming static so they can finally hear the actual game being played.

Technical Summary: naPINN – Noise-Adaptive Physics-Informed Neural Networks

1. Problem Statement

Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving inverse problems and discovering governing equations from observational data. However, their performance is highly sensitive to corrupted measurement data. In real-world scenarios, measurements are often sparse, and data can be contaminated by sensor faults, transmission errors, or environmental interference, leading to non-Gaussian, asymmetric, heavy-tailed noise, and gross outliers.

Standard PINN formulations rely on Mean Squared Error (MSE) loss, which corresponds to a Gaussian noise assumption. Under complex corruption, a small fraction of outliers can dominate the gradients, biasing the recovered solution and physical parameters. While existing robust PINN variants (e.g., using $L_1$ loss or $q$ -Gaussian likelihoods) offer some improvement, they typically rely on fixed penalty functions or pre-specified likelihood families. Crucially, they do not infer measurement reliability directly from the data residuals under unknown noise distributions, nor do they perform adaptive, sample-level selection during training.

2. Methodology: naPINN Framework

The authors propose naPINN (Noise-Adaptive Physics-Informed Neural Networks), a modular framework designed to recover physical solutions from corrupted measurements without prior knowledge of the noise distribution. The core mechanism involves estimating the reliability of measurement residuals and using this signal to adaptively regulate the influence of individual data points.

2.1 Core Components

The framework consists of three primary components working in tandem:

Residual-Based Noise Distribution Estimator: A module (instantiated primarily as a one-dimensional Energy-Based Model, EBM, in the paper) that learns the probability density of normalized residuals. It converts residuals into reliability scores, where low-probability residuals (outliers) receive low reliability scores.
Trainable Reliability Gate: A gating mechanism that converts the reliability scores into adaptive weights ( $g_i$ ) for each measurement. Measurements with anomalous behavior (low reliability) are downweighted in the data loss calculation.
Rejection-Cost Regularizer: A regularization term ( $L_{rej}$ ) that penalizes the model for discarding valid data, preventing the trivial solution where the gate rejects all measurements to minimize loss.

2.2 Training Procedure

The training process follows a three-stage protocol (Algorithm 1):

Phase 1: Warm-up: The PINN and PDE parameters are trained using the standard composite loss (physics + data) without gating. This allows the predictor to align reasonably with the data, making subsequent residuals more informative about corruption.
Phase 2: Estimator Initialization: Measurement residuals are computed from the warmed-up predictor. These residuals are normalized using a running standard deviation (updated via Exponential Moving Average) to handle scale variations. The noise distribution estimator is then initialized or fitted on these normalized residuals.
Phase 3: Joint Optimization: The PINN, the noise estimator, and the reliability gate parameters are jointly optimized. The total loss function is:
$L_{total} = L_{PDE} + L_{gate}^d + \lambda_{rej} L_{rej}$
Where $L_{gate}^d$ is the data loss weighted by the reliability gate, and $L_{rej}$ is the rejection cost. As the physical predictor improves, the residual reliability becomes a sharper proxy for corruption, allowing the gate to effectively suppress outliers.

2.3 Probabilistic Interpretation

The authors note that the gated objective admits a probabilistic interpretation similar to a latent-variable robust likelihood. The gate weight $g_i$ acts as the posterior probability that an observation is "clean," derived from the learned noise density and a learned cutoff threshold, rather than a hand-specified outlier distribution.

3. Key Contributions

The paper outlines three main contributions:

Problem Identification: The authors identify measurement-driven inverse PINNs under unknown noise distributions and gross outliers as a critical but insufficiently addressed setting for robust physics recovery.
Framework Proposal: They propose naPINN, a modular framework that couples residual-based noise distribution estimation with adaptive measurement selection via a reliability gate and rejection-cost regularization. The main instantiation uses an EBM, though the framework supports other density estimators (e.g., KDE, GMM).
Empirical Validation: The framework is validated on 2D PDE benchmarks (Allen-Cahn, Burgers, and $\lambda$ - $\omega$ Reaction-Diffusion) under multimodal non-Gaussian noise and varying outlier ratios (5% to 15%).

4. Experimental Results

The authors evaluated naPINN against standard PINNs and robust baselines (B-PINN, LAD-PINN, OrPINN) across three PDE benchmarks.

Performance: naPINN consistently achieved the lowest relative Mean Absolute Error (rMAE) and relative Root Mean Squared Error (rMSE) across all outlier ratios. For instance, on the Burgers equation with 15% outliers, naPINN improved rMAE by 58.86% over the second-best baseline (OrPINN with $q=2.9$ ).
Robustness: Unlike standard PINNs, whose performance degraded rapidly with increasing outlier ratios, naPINN exhibited only marginal degradation. Its error rates remained close to those obtained under clean measurement conditions.
Qualitative Analysis: Visualizations of solution reconstruction showed that while standard PINNs produced significant errors near sharp transitions (where outliers were concentrated), naPINN maintained accuracy comparable to the ground truth.
Outlier Detection: The learned gate successfully identified and suppressed outliers. Confusion matrix analysis on the Allen-Cahn benchmark (10% outliers) showed near-perfect precision and recall in distinguishing outliers from normal noisy measurements.
Noise Modeling: The EBM component was shown to accurately capture the complex, multimodal structure of the injected noise distribution after joint training, whereas it failed to do so at initialization.

5. Significance and Limitations

Significance:
The paper claims that naPINN is the first inverse-PINN framework to address corrupted measurements with unknown noise distributions by coupling residual-based noise estimation with trainable per-measurement reliability gating. It demonstrates that learning the noise structure and adaptively filtering data simultaneously allows for robust physics recovery even under severe data corruption. Furthermore, the framework provides interpretable insights into noise structure and can function as an anomaly detection tool for sensor failures.

Limitations:
The authors modestly acknowledge that the evaluation is based on controlled simulated benchmarks with injected corruptions. While this protocol allows for fair stress testing with known ground truth, it does not replace validation on deployment-scale real-world sensor data. Future work is suggested to evaluate naPINN on real instrumented systems, addressing challenges such as spatially correlated sensor failures and temporal drift.