⚡ electrical engineering

Impact of Physics-Informed Features on Neural Network Complexity for Li-ion Battery Voltage Prediction in Electric Vertical Takeoff and Landing Aircrafts

This paper demonstrates that integrating physics-based features into neural networks allows for significantly simpler, more efficient models that maintain high accuracy for Li-ion battery voltage prediction in eVTOL applications.

Original authors: Eymen Ipek, Mario Hirz

Published 2026-02-10

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Eymen Ipek, Mario Hirz

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 predict how much "juice" is left in a high-performance battery used in an electric flying taxi (an eVTOL).

This paper explores two different ways to teach that robot, and one way is much smarter than the other.

The Two Students: The "Pure Guesswork" Student vs. The "Smart Assistant" Student

To understand the research, imagine you are teaching a student how to predict the weather.

1. The Pure Data-Driven Student (The FNN):
This student has no idea how science works. They have never heard of "pressure," "temperature," or "wind." Instead, they just stare at millions of past weather reports. They look for patterns like, "Every time the sky looks this shade of grey, it rains two hours later."

The Problem: Because they don't understand why things happen, they need a massive, heavy textbook (a huge, complex Neural Network) to memorize every single tiny detail. If a weird storm comes along that wasn't in their textbook, they panic and make huge mistakes. They are "heavy" and slow to think.

2. The Physics-Informed Student (The PINN):
This student is different. They have a basic science textbook that explains the fundamental laws of weather. They don't try to memorize every single raindrop; instead, they use the science formulas to make a "smart guess" first. Then, they use their brain only to figure out the tiny errors that the formulas missed.

The Benefit: Because they already understand the "rules of the game," they don't need a massive textbook. They can be much "lighter" and faster, yet they are actually more accurate because they aren't just guessing—they are applying logic.

What the Researchers Actually Did

In the world of flying electric taxis, batteries are pushed to the absolute limit. They have to provide massive bursts of power to lift the aircraft off the ground, which causes the voltage to behave wildly.

The researchers compared these two "students" using real battery data:

The FNN (The Pure Guesswork approach): They built a massive, complex digital brain that tried to learn everything from scratch. It was heavy, used a lot of computer memory, and struggled with accuracy.
The PINN (The Physics-Informed approach): They gave the digital brain a "cheat sheet" based on an Equivalent Circuit Model (ECM)—a mathematical recipe that describes how electricity naturally flows through a battery. The brain only had to learn the "leftover" bits that the recipe couldn't explain.

The "Aha!" Moment (The Results)

The results were a landslide victory for the "Smart Assistant" (the PINN):

Lighter and Faster: The physics-informed model used 75% fewer "brain cells" (parameters) than the complex data-only model. This is huge because flying taxis have small, limited computers on board; they don't have room for a massive, heavy "brain."
More Accurate: Even though it was much smaller, the physics-informed model was much better at predicting voltage. It cut errors (like the "oops" moments in prediction) by about 50%.
Better Under Pressure: When the battery was working hard during takeoff or landing, the physics-informed model stayed steady, while the pure data model struggled to keep up.

Why This Matters for the Future

If we want electric flying taxis to be safe, we need to know exactly how much power is left in the battery at every second.

This paper proves that we don't need to build giant, power-hungry supercomputers to do this. By giving AI a little bit of "common sense" (physics), we can make it smaller, faster, and much more reliable—perfect for the tiny computers tucked away inside a flying vehicle.

Technical Summary: Impact of Physics-Informed Features on Neural Network Complexity for Li-ion Battery Voltage Prediction

1. Problem Statement

The electrification of electric Vertical Takeoff and Landing (eVTOL) aircraft introduces extreme operational challenges for Battery Management Systems (BMS). eVTOL mission profiles involve aggressive power dynamics (high C-rates) during takeoff and landing, causing rapid voltage drops and thermal stress.

Current modeling approaches face a fundamental trade-off:

Physics-based models (e.g., P2D): Highly accurate but computationally too heavy for real-time, on-board implementation.
Equivalent Circuit Models (ECMs): Computationally efficient but struggle to capture the non-linear electrochemical behaviors and transient dynamics caused by high-power pulses.
Purely Data-Driven Models (e.g., DNNs): Capable of capturing non-linearities but act as "black boxes," require massive datasets, possess high computational/memory overhead, and suffer from poor extrapolation outside training domains.

2. Methodology

The authors propose a Physics-Informed Machine Learning (PIML) approach using a hybrid residual model. Instead of forcing a neural network to learn the entire voltage response from scratch, the physics-based dynamics are used to provide a "baseline" prediction.

The Physics Prior: A second-order Equivalent Circuit Model (2RC) is used to calculate a physics-based voltage guess ( $V_{phy}$ ). This model accounts for the Open Circuit Voltage (OCV), internal resistance ( $R_0$ ), and two RC polarization states ( $V_{RC1}, V_{RC2}$ ).
The Neural Network Task: The neural network is tasked only with learning the residual error ( $\Delta V_\theta$ )—the difference between the physics-based guess and the actual measured voltage.
Model Formulation: The final prediction is $V_{pred} = V_{phy} + \Delta V_\theta(x)$ .
Data & Training: The study utilizes an open-source eVTOL battery dataset. The models were trained on four battery cells (representing various thermal and mission profiles) and tested on a fifth, unseen cell (VAH11) to evaluate generalization.
Comparison Framework: The researchers compared standard Feedforward Neural Networks (FNN) against the proposed Physics-Informed Neural Networks (PINN) across varying architectures (layers and neurons).

3. Key Contributions

Hybrid Residual Architecture: Integration of a 2RC model as a structural prior within a neural network framework.
Feature Augmentation: The PINN uses an augmented input vector that includes not just raw data (Current, SOC, Temp), but also physics-derived terms ( $V_{RC}$ states and $V_{phy}$ ), significantly narrowing the hypothesis space the network must search.
Complexity-Fidelity Analysis: A systematic quantification of how much "intelligence" can be offloaded to physics to reduce the required size (parameters) of the neural network.

4. Results

The results demonstrate that the PINN approach is both more accurate and more efficient than the FNN approach:

Accuracy: The optimal PINN configuration (L2-N64) achieved an RMSE of 20.1 mV and an $R^2$ of 99.2%. In contrast, even the most complex FNN tested (L4-N128) only reached an RMSE of 39.2 mV.
Error Reduction: The PINN reduced both RMSE and peak transient errors by approximately 50% compared to pure data-driven models.
Model Compression: The PINN achieved superior performance while using up to 75% fewer trainable parameters than complex FNNs. For example, a minimal single-layer PINN (L1-N32) outperformed a much larger FNN.

5. Significance

This research is highly significant for the deployment of Edge AI in aviation. By proving that physics-informed features allow for drastic model compression without sacrificing fidelity, the authors provide a pathway for implementing high-accuracy voltage prediction on resource-constrained, on-board BMS hardware. This enables safer and more precise State-of-Charge (SOC) and State-of-Power (SOP) estimation, which are critical for the flight safety of eVTOL aircraft.