Physics-Informed Neural Networks (PINNs) are an innovative class of deep neural network models that directly embed the laws of physics—typically formulated as partial or ordinary differential equations (PDEs or ODEs)—into the training process. Instead of relying solely on large volumes of data, PINNs augment the traditional data-driven loss with additional terms that penalise deviations from established physical laws. In practice, this means that during training, the network isn’t just learning to fit observed data; it’s also learning to satisfy the underlying equations that govern the phenomena in question, such as conservation of mass, momentum, or energy.

At the heart of this approach lies the idea of using automatic differentiation (a key feature in modern machine learning frameworks) to compute derivatives of the neural network’s output concerning its inputs. These derivatives are substituted into the physical equations (for example, the Navier–Stokes equations in fluid dynamics or the heat equation in thermal processes) to form a residual. The network is then trained to minimise both the difference between its predictions and any available data and this residual error, effectively “informing” the network of the physics behind the problem. This dual focus—balancing data fidelity with adherence to physical laws—often leads to solutions that generalise well, even when experimental or observational data are sparse or noisy.

The power of PINNs is particularly evident in applications where classical numerical methods (like finite element or finite volume methods) struggle, such as in domains with complex geometries or in inverse problems where parameters of the system must be inferred from limited measurements. By integrating physical constraints, PINNs narrow the solution space to physically plausible outcomes, which can lead to more robust predictions and a deeper understanding of the system under study. However, they also present new challenges: the relative weighting of data-driven loss components versus the physics-based constraints must be carefully balanced, and optimising such combined loss functions can be computationally demanding. Ongoing research is actively addressing these challenges to extend their applicability to increasingly complex, high-dimensional systems.

PINNs exemplify how blending data science with domain-specific physical knowledge can lead to powerful hybrid models—models capable of solving forward simulations, parameter estimation, and even data assimilation tasks while honouring the immutable laws of nature.