AI Technologies for Solving Ordinary Differential Equations

Explore how deep learning uses artificial neural networks with weights and biases, activation functions like ReLU and sigmoid, and loss-driven training optimized by gradient descent and backpropagation.

Learn how PINNs, or physics-informed neural networks, solve ordinary differential equations. Train with boundary and initial conditions, physics losses, domain points, and a standard optimizer to enforce the governing equation.

Solve the harmonic and damped oscillator equations using the exact solution, with initial conditions and a 0–1 time span in 300 steps, to compare with pins results.

Define and implement a one-input, one-output feed-forward neural network in PyTorch to map time to position, configuring a single hidden layer with tanh activation and a forward pass.

Balance three losses in the model: initial condition for X at time zero, velocity DU/DT, and the physics loss from the equation, by tuning lambda1 and lambda2.

Learn how to evaluate results of physics-informed neural networks by plotting training progress against the exact solution, comparing the network solution, and assessing losses, training steps, and inference speed.

Change the loss function from absolute to squared to accelerate training and demonstrate how different loss definitions, aligned with physics, affect convergence in neural network solutions of differential equations.

Demonstrates solving a differential equation with a pins network inside torch through a forward feed neural net, enforcing initial conditions, and training with physics loss to match the exact solution.

Explore inverse pins for solving a damped harmonic oscillator, estimating the unknown mu from data by adapting a pins network and comparing with standard pins.

Create data for inverse pinns by sampling the exact solution, generate 40 observations with torch, and use these samples to predict mu in an undamped pendulum.

We define the IPINNS network, identical to pins, add a trainable mu parameter, and train the model with physics loss at t-physics points, boundary data, and observation data.

Demonstrate training a pins network to fit observed points, using plots every 5,000 steps, detaching gradients, and restarting memory to improve accuracy.

Train, plot, and visualize the mu parameter to obtain a final value of 4, framing it as an inverse problem to fit data and recover mu.

Preprocess data for deep onets by reshaping the full u sequence and a single x location to feed branch and trunk nets, producing 11,000 data points.

Convert raw data into tensors and patches with a 32-patch size, split into train/test, and implement PyTorch data loaders and datasets to prepare inputs for the model.

Define a mean squared error loss and set up an optimizer to train a neural network, then run a 2000-step training loop with back propagation and progress monitoring.

Observe how a deep neural operator learns to integrate a cosine function as the analytic solution guides evaluation, demonstrating loss convergence and close agreement with the 1/5 sin(5x) result.