Automatic differentiation (AD) is the silent engine behind neural network training in PyTorch, but when it comes to Physics-Informed Neural Networks (PINNs), the challenge multiplies. In these models, the loss depends not only on the network output but also on its derivatives with respect to the input — the so-called 'physics derivatives' — which forces the construction of a computational graph on top of another graph. This article takes a technical and business perspective on how PyTorch handles this double differentiation, and how your organization can leverage this knowledge through custom software applications that integrate artificial intelligence, cybersecurity, cloud, and business intelligence.
A typical PINN solves ordinary or partial differential equations. For instance, if we train a 1-3-3-1 multilayer perceptron to solve y''(t) + y(t) = 0 with initial condition y(0)=1, the loss function includes the equation residual, which in turn depends on the second time derivative of the network output. PyTorch, through its autograd engine, builds a computational graph during the forward pass that records all operations. When the physics derivative \(\hat{y}''(t)\) is needed, a partial backward pass with create_graph=True is executed, generating a second graph that later allows computing parameter gradients with respect to the total loss. This 'graph-on-graph' mechanism is essential and consumes more memory, but it is the only way to obtain exact gradients for stable PINN training.
In practice, companies wanting to implement PINNs for process simulation, engineering design, or physical system modeling face several challenges: scaling training in the cloud, protecting sensitive data generated during simulations, and extracting business insights from results. This is where Q2BSTUDIO, as a software and technology development company, offers comprehensive solutions. For example, by combining AI with our AWS and Azure cloud capabilities, we can deploy distributed training pipelines for PINNs that reduce computation time from weeks to hours. Additionally, we integrate cybersecurity from the design phase to protect both models and data, and we use Power BI to visualize predictions and residuals in interactive dashboards that facilitate decision-making.
The automatic differentiation architecture in PyTorch is based on adjoints: during the backward pass, each node receives an error signal that propagates backward using vector-Jacobian products. In PINNs, when the physics derivative graph is built with create_graph=True, a new graph is created whose nodes are differentiation operations. This allows the final backward pass — the one that computes parameter gradients — to traverse both differentiation levels. Each of the 22 parameters in the example (weights and biases of a 1-3-3-1 network) receives its exact gradient in a single reverse pass. Without this mechanism, one would have to resort to numerical differentiation, which is unstable and costly.
For a business, adopting PINNs means investing in infrastructure and talent. But with the right approach, the benefits are enormous: real-time fluid simulation, industrial process optimization, mechanical failure prediction, and more. Q2BSTUDIO helps organizations make this leap by developing custom applications that integrate AI agents capable of automatically tuning model hyperparameters, or that connect with BI systems to feed dashboards with physical predictions. Furthermore, we ensure that the entire model lifecycle — from local development to cloud deployment — meets the highest cybersecurity standards.
Understanding how automatic differentiation works in PyTorch for PINNs is not just an academic exercise: it is the foundation for building robust and scalable software. Whether you need an AI solution for simulation, a physics-based process automation system, or simply want to explore how cloud and BI can boost your models, at Q2BSTUDIO we have the technical expertise and business vision to make it happen. We invite you to contact us to discuss how we can help you implement these technologies in your organization.
