⚡ electrical engineering

A data-driven model-free physical-informed deep operator network for solving nonlinear dynamic system

This paper proposes a data-driven, model-free Deep Operator Network framework that integrates a surrogate machine learning model to capture short-term dependencies from limited experimental data, enabling the effective approximation of nonlinear dynamic systems without requiring exact mathematical formulas or vast datasets.

Original authors: Jieming Sun, Lichun Li

Published 2026-02-24

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CC BY 4.0

Original authors: Jieming Sun, Lichun Li

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

Imagine you are trying to teach a robot how to drive a car, but you have a major problem: you don't have the owner's manual (the physics equations), and you only have a few hours of video footage (limited data) to learn from.

Most advanced AI systems today need either the complete owner's manual or thousands of hours of driving video to learn how to drive. If you give them just a little bit of video and no manual, they usually crash.

This paper introduces a clever new way to teach the robot to drive using a "hybrid" approach. Here is the breakdown in simple terms:

1. The Problem: The "Blind" and the "Starved"

The "Blind" (No Physics): In the real world, we often don't know the exact math formulas that govern a system (like a swinging pendulum or a chaotic weather pattern).
The "Starved" (No Data): We also can't always run thousands of expensive experiments to get huge datasets. We might only have a few test runs.
The Result: Standard AI models fail because they are either too hungry for data or too confused without the rules.

2. The Solution: The "Short-Term Memory" Trick

The authors realized that even if you don't know the whole story of how a system works, you usually know a tiny piece of it: what happens right now depends heavily on what happened just a second ago.

The Analogy: Imagine you are walking down a hallway. You don't need to know the architectural blueprints of the whole building to know that if you take a step forward, you will be a few inches closer to the wall. You only need to know your immediate past to predict your immediate future.
The "Surrogate" Model: The researchers built a small, simple AI (a "surrogate") just to learn this short-term memory. It learns: "If the car was going this fast a second ago, and I turned the wheel this much, it will likely be here a split second later."

3. The Masterpiece: The "DeepOnet" with a Brain Implant

They then took a powerful, complex AI called DeepOnet (which is great at learning patterns but needs lots of data) and gave it a "brain implant."

The Setup:
- The DeepOnet is the main driver. It looks at the starting conditions and the steering commands to predict where the car will be in the future.
- The Surrogate Model is the "Physics Coach." It doesn't drive the car; it just whispers to the DeepOnet: "Hey, based on the laws of motion we know, if you predict the car will be there, that's impossible because it would have to teleport!"
How it works together: The DeepOnet tries to guess the future. The "Physics Coach" checks if that guess makes sense based on the short-term rules it learned. If the guess is wrong, the DeepOnet gets a "scolding" (a penalty in its training) and tries again.

4. The Results: Learning with Less

They tested this on three different "cars":

A Swinging Pendulum: Like a child on a swing.
A Driven Oscillator: Like a spring bouncing up and down.
A Chaotic Jerk System: Like a rollercoaster that goes crazy and unpredictable.

The Outcome:

Standard AI (DeepOnet alone): With limited data, it got lost. It predicted the swing would stop or go the wrong way.
Their New Method: Even with very little data and some "static" (noise) in the measurements, their hybrid model nailed the predictions. It was like a driver who, despite not having the manual, learned to drive perfectly by paying close attention to the immediate road conditions and listening to a wise coach.

The Big Takeaway

This paper is about making AI smarter with less.

Instead of demanding a massive library of data or a perfect textbook of physics, this method teaches AI to use its short-term memory to fill in the gaps. It's a bit like teaching a child to ride a bike: you don't need to explain the complex physics of angular momentum; you just teach them to balance based on what they feel right now, and they figure out the rest.

This is a huge step forward for engineering fields where experiments are expensive and the math is too complicated to write down perfectly.

1. Problem Statement

The paper addresses a critical bottleneck in the analysis of nonlinear dynamic systems: the difficulty of performing accurate state estimation when both precise mathematical models (governing equations) and large-scale experimental datasets are unavailable.

Limitations of Existing Methods:
- Physics-Informed Neural Networks (PINNs): Require exact knowledge of the governing physical laws (e.g., specific Partial Differential Equations) to formulate the loss function. They fail when the mathematical formulation is unknown or incomplete.
- Standard Deep Operator Networks (DeepOnet): Are model-free but rely heavily on massive amounts of training data to learn the mapping between input functions and output states. They struggle in data-scarce environments.
The Challenge: In many real-world engineering scenarios, researchers possess only a few experimental data points and limited system knowledge (e.g., knowing the order of the system's differential equation but not the specific coefficients or functional forms).

2. Methodology

The authors propose a novel framework: a Data-Driven, Model-Free, Physics-Informed Deep Operator Network. The core innovation is replacing the explicit physical law (usually required by PINNs) with a learned short-term dependency derived from limited data.

The methodology proceeds in three main stages:

A. Theoretical Foundation: Short-Term Dependency

Based on the Euler method for discretizing continuous-time systems, the authors invoke Theorem 2.1, which posits that for an $n$ -th order nonlinear dynamic system, the current state $x(t)$ depends only on the preceding $n$ steps of states and inputs ( $x(t-n) \dots x(t-1)$ and $u(t-n) \dots u(t-1)$ ), provided the sampling period is sufficiently small.

Instead of using the unknown function $F(\cdot)$ in the ODE, the system's "physics" is approximated by this temporal dependency.

B. Step 1: Surrogate Model for Short-Term Dependency

A surrogate machine learning model (specifically a Feed-Forward Neural Network, FFNN) is trained on the limited available data to learn the mapping $\gamma$ .

Inputs: Past states ( $x(t-n) \dots x(t-1)$ ), past derivatives, and past control inputs.
Output: The highest-order derivative or the current state $x(t)$ .
Purpose: This FFNN acts as a "physics-informed" component that captures the intrinsic dynamic rules of the system without needing the explicit ODE formula.

C. Step 2: Integration into DeepOnet

The trained FFNN is integrated into the DeepOnet architecture to form the proposed hybrid model:

DeepOnet Structure:
- Branch Net: Encodes the initial conditions and the sequence of control inputs ( $u(0) \dots u(t)$ ).
- Trunk Net: Encodes the time coordinate ( $t$ ).
Hybrid Loss Function: The total loss $L$ $L$ combines:
1. Data Loss ( $L_{data}$ ): Mean Squared Error (MSE) between the DeepOnet prediction and the sparse experimental data.
2. Physics-Informed Loss ( $L_P$ ): The discrepancy between the derivative predicted by the DeepOnet (via automatic differentiation) and the derivative predicted by the surrogate FFNN.
- Note: This forces the DeepOnet to adhere to the short-term dynamic rules learned by the FFNN, effectively acting as a constraint similar to a physical law.

3. Key Contributions

Model-Free Physics-Informed Learning: The paper introduces a method to incorporate "physics" into DeepOnet without requiring the explicit mathematical formula of the system. The "physics" is learned directly from data via the short-term dependency surrogate.
Data Efficiency: The framework is specifically designed to work with sparse datasets (as few as 5 training sets in the experiments), overcoming the data-hungry nature of standard DeepOnets.
Robustness to Noise: The method demonstrates superior performance in the presence of measurement noise compared to standard DeepOnet, likely due to the regularization effect of the learned short-term dependency.
Generalizability: The approach is validated across systems of varying complexity (2nd-order and 3rd-order ODEs) and stability characteristics (stable, marginally unstable, and chaotic).

4. Experimental Results

The authors evaluated their method on three distinct nonlinear dynamic systems, comparing it against a standard DeepOnet benchmark.

System	Type	Order	Data Condition	Metric (MAE) - Proposed	Metric (MAE) - Standard DeepOnet	Outcome
Pendulum	Nonlinear (Stable)	2nd	Noise-Free	6.23	17.98	Proposed method significantly more accurate.
Pendulum	Nonlinear (Stable)	2nd	Noisy ( $\sigma=0.002$ )	7.57	19.27	Proposed method maintains accuracy; DeepOnet degrades.
Driven Oscillator	Linear/Nonlinear (Unstable)	2nd	Noise-Free	16.69	36.22	Proposed method captures trends better.
Driven Oscillator	Linear/Nonlinear (Unstable)	2nd	Noisy ( $\sigma=0.01$ )	19.10	39.46	Proposed method is nearly 2x more accurate.
Chaotic Jerk	Chaotic	3rd	Noise-Free	19.8	10.5*	Note: In this specific noise-free case, DeepOnet had lower MAE, but the text notes the proposed method showed "enhanced accuracy" in visual trends and robustness in noisy conditions (MAE 18.7 vs 22.2).
Chaotic Jerk	Chaotic	3rd	Noisy ( $\sigma=0.03$ )	18.7	22.2	Proposed method outperformed in noisy conditions.

Key Finding: In all noisy scenarios, the proposed method consistently outperformed the standard DeepOnet, demonstrating that the surrogate model effectively regularizes the learning process against data imperfections.

5. Significance

This research bridges a critical gap in scientific machine learning (SciML). By decoupling the need for explicit physical equations from the need for massive datasets, the proposed framework enables:

Offline State Estimation: Accurate prediction of system states in scenarios where sensors are limited or expensive.
Application to Unknown Systems: The ability to model complex, nonlinear, or chaotic systems where the governing equations are too difficult to derive analytically.
Practical Engineering Utility: The method offers a robust tool for real-world applications (e.g., structural health monitoring, robotics, climate modeling) where data is often noisy and incomplete, and exact models are unavailable.

In summary, the paper successfully demonstrates that learning the temporal structure (short-term dependency) of a system can serve as a surrogate for physical laws, enabling Deep Operator Networks to function effectively in data-scarce, model-unknown environments.