Understanding Physics-Informed Neural Networks (PINNs) — Part 1
Physics-Informed Neural Networks (PINNs) represent a unique approach to solving problems governed by Partial Differential Equations (PDEs). By combining the universal function approximation capabilities of neural networks with the precision of physical laws, PINNs enable the resolution of complex systems where traditional methods often fall short. In this post, inspired by the paper by Raissi et al., we will systematically break down every key concept as a introduction to PINNs
Differential Equations: The Foundation
Differential equations are mathematical models that describe the relationship between changing quantities and their rates of change. These equations are fundamental in physics, engineering, and other sciences.
Ordinary vs. Partial Differential Equations
Ordinary Differential Equations (ODEs): ODEs involve derivatives with respect to a single variable, typically time t . For instance:
where y(t) is the state variable, and k is a constant. ODEs model one-dimensional systems like population growth, radioactive decay, etc.
Partial Differential Equations (PDEs): PDEs involve derivatives with respect to multiple variables, such as time t and space x . For example:
where u(t, x) could represent the temperature in a material. PDEs govern phenomena like heat transfer, fluid flow, wave mechanics, etc.
Why Are PDEs Critical?
PDEs underpin the mathematical models for real-world phenomena:
- Heat diffusion (heat equation).
- Fluid dynamics (Navier-Stokes equations).
- Quantum mechanics (Schrödinger equation).
Despite their significance, solving PDEs analytically is often infeasible for complex systems. Numerical methods like finite difference or finite element methods are commonly used but face challenges such as:
1. High Computational Cost: Especially in multi-dimensional systems.
2. Discretization Errors: Arising from approximations over grids.
3. Scalability Issues: Limited applicability to problems in high dimensions or with irregular boundaries.
PINNs address these challenges by approximating solutions with neural networks, embedding the physics directly into their structure.
What Are Physics-Informed Neural Networks: Introducing PINNs
PINNs leverage neural networks to approximate solutions to PDEs while incorporating physical laws as constraints. Instead of relying purely on data to learn relationships, PINNs leverage the structure of known physics to guide their solutions. This hybrid approach balances data and physics, ensuring robustness and accuracy, even with sparse or noisy data.
What Is a PDE Residual?
A key concept in PINNs is the PDE residual, which measures how well a candidate solution satisfies the governing equations.
Example:
For the heat equation:
the residual is defined as:
Here, u(t, x; θ) is the neural network approximation with parameters θ. Ideally, f(t, x; θ) = 0 , indicating that the approximation satisfies the PDE. During training, the network minimizes the residual across the domain, progressively refining its approximation.
Loss Function Design
PINNs use a composite loss function that combines data consistency and physics adherence.
1. Data Loss
Penalizes errors between network predictions and observed data:
where (t_i, x_i, u_i) are data points.
2. Physics Loss:
Penalizes deviations from the PDE residual at collocation points:
The total loss:
ensures that the solution adheres to both the data and the physics.
What Are Collocation Points?
Collocation points are randomly sampled points in the problem domain where the PDE residual is evaluated. They ensure that the neural network respects the physics globally, not just at observed data points.
For example, in a rod with length L = 10 and time interval T = 1 , collocation points might be randomly sampled in [0, 1] x [0, 10] .
Training a PINN: Optimization and Automatic Differentiation
Training involves minimizing the total loss with respect to the network parameters θ. This process ensures that the solution satisfies both the data and the physics.
PINNs rely on automatic differentiation (AD) to compute derivatives from the neural network. Automatic Differentiation computes derivatives using computational graphs, ensuring high precision, avoiding errors associated with numerical differentiation.
Optimization Algorithms:
- Adam: An adaptive gradient-based method effective in early training stages.
- L-BFGS: A quasi-Newton method that converges quickly near the optimal solution. It is well-suited for smaller datasets.
Example: Schrödinger Equation
If you are still reading, here is your 1 minute of Physics 101.
The Schrödinger equation is a cornerstone of quantum mechanics, describing how a particle’s wavefunction evolves over time. This wavefunction contains all the information about the system, including its position, momentum, and energy probabilities. Solving this equation accurately is essential in many fields, including physics, chemistry, and material science.
Physics-Informed Neural Networks (PINNs) offer a modern approach to solving the Schrödinger equation. By embedding the equation’s physical laws into a neural network, PINNs reduce reliance on large datasets, improve accuracy, and handle complex scenarios that challenge traditional methods.
The Schrödinger equation describes the evolution of a wave function ψ(t, x) as:
where:
- ψ(t, x) : The complex-valued wave function.
- i: Imaginary unit ( i² = -1 ).
- ψ_t: Time derivative of ψ.
- ψ_xx: Second spatial derivative of ψ .
- |ψ|² : Magnitude squared of ψ , introducing nonlinearity
The equation contains:
1. Time Evolution: Governs how the wave function evolves over time.
2. Dispersion: Captures spatial spreading of the wave function.
3. Nonlinear Term: Represents interactions within the system.
Decomposing into Real and Imaginary Parts
The wavefunction ψ(t, x) is complex-valued. To work with real-valued neural networks, we decompose ψ(t, x) into:
where:
- u(t, x) : Real part of ψ.
- v(t, x) : Imaginary part of ψ .
Substituting this into the Schrödinger equation gives:
Separating real and imaginary parts leads to two coupled equations:
Real part:
Imaginary part:
Defining the Residuals
For PINNs, we define the residuals for the real and imaginary parts, respectively:
These residuals measure how well the neural network solution u(t, x; θ) and v(t, x; θ) satisfies the Schrödinger equation.
PINN Framework for the Schrödinger Equation
We use two neural networks to approximate u(t, x) and v(t, x) : u(t, x; θ) and v(t, x; θ),
where θ represents the network parameters (weights and biases). The network inputs are t (time) and x (space), and the outputs are the real and imaginary components of the wavefunction.
Loss Function
The total loss function combines:
1. Data Loss
2. Physics Loss
The total loss:
Initial and Boundary Conditions
A typical test case for the Schrödinger equation is soliton propagation, where the wavefunction maintains its shape during evolution. The initial condition is:
where:
• A : Amplitude of the wave.
• B : Wave width.
• C : Phase constant.
Boundary conditions are often periodic:
Training the PINN
Derivatives are computed using automatic differentiation, avoiding numerical errors. Residuals are evaluated at N_f collocation points sampled randomly in the (t, x) domain. The total loss is minimized using: Adam Optimizer for initial training and L-BFGS for fine-tuning near convergence.
Key Takeaways
- By embedding the Schrödinger equation into the loss function, PINNs enforce physical constraints on the solution, ensuring adherence to the laws of quantum mechanics.
- The nonlinear term | ψ|² ψ is naturally incorporated into the framework, enabling PINNs to solve challenging problems like soliton propagation.
- PINNs reduce reliance on data by leveraging physical laws, making them effective even in data-scarce scenarios.
- The neural network provides a continuous, differentiable approximation of ψ(t, x) across the entire domain, without requiring grid-based discretization.
Conclusion: A Starting Point for Understanding Physics-Informed Neural Networks
Physics-Informed Neural Networks (PINNs) represent a significant step forward in solving problems governed by physical laws, seamlessly integrating the mathematical rigor of differential equations with the adaptability of neural networks. By embedding physics directly into the training process, PINNs offer a unique blend of data-driven learning and physics-constrained modeling.
In this post, we began by laying a solid foundation, starting from the fundamental differences between ordinary and partial differential equations, the importance of PDEs in modeling complex systems, and why traditional methods often struggle. We then explored how PINNs address these challenges, focusing on key concepts like PDE residuals, loss function design, collocation points, and optimization strategies.
The detailed walkthrough of solving the Schrödinger equation showcased how PINNs operate in practice. By splitting the wavefunction into real and imaginary parts, defining physics-driven residuals, and leveraging automatic differentiation, PINNs provide accurate, continuous, and scalable solutions to nonlinear and complex problems. This example serves as a template for understanding how PINNs can be applied to other challenging equations in science and engineering.
Looking Ahead
In the coming posts, I will continue breaking down key papers, exploring foundational concepts, advanced methodologies, and their practical implementations.
The goal is to create a clear, accessible roadmap through the growing body of PINN research, helping us bridge the gap between traditional computational science and cutting-edge machine learning techniques.
If you’ve enjoyed this post or have specific questions or topics you’d like to see covered, let me know. This journey is personally very exciting to me, and this blog post series will take a step by step approach.
