PINTO: Physics-informed transformer neural operator for learning generalized solutions of partial differential equations for any initial and boundary condition

Applications in physics, engineering, mechanics, and fluid dynamics necessitate solving nonlinear partial differential equations (PDEs) with different initial and boundary conditions. Operator learning, an emerging field, solves these PDEs by employing neural networks to map the infinite-dimensional input and output function spaces. These neural operators are trained using data (observations or simulations) and PDE residuals (physics loss). A key limitation of current neural methods is the need to retrain for new initial/boundary conditions and the substantial simulation data required for training. We introduce a physics-informed transformer neural operator (named PINTO) that generalizes efficiently to new conditions, trained solely with physics loss in a simulation-free setting. Our core innovation is the development of iterative kernel integral operator units that use cross-attention to transform domain points of PDE solutions into initial/boundary condition-aware representation vectors, supporting efficient and generalizable learning. The working of PINTO is demonstrated by simulating important 1D and 2D equations used in fluid mechanics, physics and engineering applications: advection, Burgers, and steady and unsteady Navier-Stokes equations (three flow scenarios). We show that under challenging unseen conditions, the relative errors compared to analytical or numerical (finite difference and volume) solutions are low, merely 20% to 33% of those obtained by other leading physics-informed neural operator methods. Furthermore, PINTO accurately solves advection and Burgers equations at time steps not present in the training points, an ability absent for other neural operators. The code is accessible at https://github.com/quest-lab-iisc/PINTO.