pino

A neural-operator approach that enforces governing physical laws during training to learn solution operators of parametric partial differential equations, producing mesh-independent, generalizable surrogates.

PINO (Physics-Informed Neural Operator) combines operator learning — frameworks that approximate mappings between function spaces (e.g., parameter fields to solution fields) — with physics-informed training objectives that embed PDE residuals, boundary/initial conditions, and conserved quantities into the loss. Unlike pointwise surrogates, neural operators such as DeepONet or the Fourier Neural Operator (FNO) learn parameter-to-solution operators directly and therefore generalize across spatial discretizations; PINO augments this by reducing or eliminating the need for dense labeled simulation data through physics-based regularization, hybrid data-physics losses, and multi-fidelity strategies. For practitioners this means faster surrogate evaluation for parametric studies, uncertainty quantification, inverse problems, and control in domains like computational fluid dynamics, climate modeling, and materials science, while retaining desirable properties such as mesh independence and better extrapolation across parameter regimes. Theoretical and practical challenges include ensuring stability and conservation, designing operator architectures and kernels that capture multi-scale behavior, managing spectral bias, and integrating boundary/initial condition encodings and solver-informed inductive biases to improve sample efficiency and robustness.

First appeared in the literature around 2020–2022 as neural-operator research matured; the term gained broader uptake across numerical PDE and ML (Machine Learning) communities between 2021 and 2023 alongside interest in FNO and physics-informed methods.