My Work with PINNs: Solving the Simple Pendulum Problem
Artwork by Madison Butchko
In my research, I am developing a Physics-Informed Neural Network (PINN) to model the dynamics of a simple pendulum—a foundational physics problem. This work is significant because it demonstrates how PINNs can handle systems governed by well-established physical laws with greater efficiency than traditional neural networks. While conventional neural networks rely heavily on large datasets, PINNs integrate physics directly into their architecture, making them ideal for situations where data is limited but physical principles are well-understood.
The pendulum problem, though seemingly simple, offers rich dynamics that span both linear and nonlinear behaviors, making it an ideal testbed for PINN applications. By embedding Newton’s second law into the neural network’s loss function, I aim to capture the pendulum's motion with minimal input data, such as its initial position and velocity. This approach not only minimizes data dependency but also ensures that the model's predictions are interpretable and consistent with physical reality.
Embedding Physical Laws into PINNs
PINNs overcome these limitations by embedding physical laws directly into the neural network’s architecture. This is done by incorporating known laws—expressed as partial differential equations (PDEs) or ordinary differential equations (ODEs)—into the model’s loss function. The loss function in a PINN thus has two components: one that minimizes prediction error from the data (when available) and another that enforces the physical law. For the pendulum problem, Newton’s second law F=ma governs the motion. By embedding this equation directly into the model, the PINN learns the system’s dynamics without needing a vast amount of data, because the physics guides the learning process.
The Steps: Building a PINN for the Simple Pendulum
Define the Problem and Equations: The first step is to clearly outline the physics problem. For the pendulum, Newton’s second law applies:
θ′′(t)+g/Lsin(θ(t))=0
where θ is the angular displacement, g is the gravitational acceleration, and L is the length of the pendulum. This second-order differential equation captures the motion of the pendulum.
Choose the Framework: PyTorch vs. TensorFlow: PINNs can be implemented using either PyTorch or TensorFlow, two of the most widely used machine learning frameworks.
PyTorch is particularly well-suited for research because of its dynamic computation graph, which allows for greater flexibility when experimenting with different architectures. Its Pythonic nature also makes it easier to debug and modify. PyTorch’s autograd feature simplifies the process of computing derivatives, a critical requirement for solving PDEs in PINNs.
TensorFlow is more commonly used for large-scale production environments. It offers robust tools for deploying models and handling large datasets. TensorFlow’s integration with Keras also provides a high-level API that can simplify PINN implementation. Additionally, TensorFlow’s automatic differentiation capabilities support the computation of gradients for the physical laws embedded in the model.
Construct the Neural Network: A basic fully connected feedforward neural network is built, where the input is time t, and the output is θ (the angular displacement of the pendulum). The network must be flexible enough to capture both linear (small angle approximation) and nonlinear dynamics (for larger swings).
Define the Loss Function: This is the key differentiator for PINNs. The loss function has two components:
Data Loss: If available, this measures the error between the network’s predictions and actual experimental data. In many cases, such as with the simple pendulum, limited data can still be used for validation.
Physics Loss: This is where the physics is embedded. Using PyTorch or TensorFlow, automatic differentiation can be used to compute derivatives of the network’s output, allowing you to enforce Newton’s second law directly. For the pendulum, this means ensuring the network outputs solutions that satisfy:
θ′′(t)+gLsin(θ(t))=0
Train the Network: Training involves minimizing both the data loss and the physics loss. By adjusting the parameters, the network learns to approximate the pendulum’s behavior in a way that adheres to the physical laws while making use of available data.
Validation and Testing: After training, the model is tested against analytical solutions of the pendulum problem or new experimental data. For instance, comparing the predicted motion with the exact solution for small-angle oscillations (where the equation simplifies) helps verify the accuracy of the model.
Why PINNs Excel
The major advantage of PINNs is that they drastically reduce the amount of data needed. For example, in the pendulum case, a traditional neural network might require data from hundreds of pendulum swings to accurately model the system. In contrast, a PINN can achieve the same level of accuracy with just a few initial conditions, because the physical laws are already embedded into the network.
Moreover, PINNs offer superior generalization. Traditional neural networks often struggle when applied to conditions outside their training data, while PINNs can generalize effectively to a broader range of scenarios. This is because the physics-based constraints guide the model even in untested conditions. For example, the pendulum model can predict behavior for angles not covered in the training set, as long as the system’s dynamics adhere to Newton’s laws.
Extending PINNs to More Complex Systems
While my current project focuses on the simple pendulum, the methodology extends to much more complex systems. PINNs are already being applied to model fluid dynamics, where the Navier-Stokes equations govern the behavior of fluids, and to quantum mechanics, where Schrödinger’s equation defines particle motion. These systems are challenging to simulate with traditional methods due to their computational complexity, but PINNs provide a new approach that leverages the best of both AI and classical physics.
The Potential of PINNs
The potential of PINNs extends far beyond solving the simple pendulum problem or fluid flow equations. Physics-Informed Neural Networks (PINNs) are redefining scientific research by merging AI’s data-driven capabilities with the rigor of physical laws. This fusion allows PINNs to solve complex problems with minimal data while providing insights into the system’s behavior—unlike traditional neural networks, which act as "black boxes." PINNs excel in fields where data is scarce, from climate science to quantum mechanics, making accurate predictions grounded in fundamental physics.
The real power of PINNs lies in their ability to generalize beyond training data. By embedding universal physical principles, PINNs can model new scenarios with greater accuracy than traditional methods. Moreover, PINNs democratize high-level modeling by reducing the need for vast datasets and computational resources. This makes advanced research accessible to smaller institutions and accelerates innovation across disciplines.
Written by Madison Butchko:
Madison Butchko is a senior at Yale University, pursuing a B.S. in Physics and a B.A. in East Asian Studies. She conducted research on Physics-Informed Neural Networks (PINNs) under Professor Sarah Beetham, focusing on computational modeling of complex physical systems. Passionate about physics and teaching, Madison plans to pursue a Ph.D. to advance her research and inspire others through education.
