Glossary

A family of methods that encodes PDE residuals into the loss function and parameterizes the solution itself with a neural network: no mesh, no labeled solution, trained purely from the governing equations and initial/boundary conditions.

A deep learning paradigm that learns a function-to-function mapping: feed in a parameter field or boundary condition and get the whole solution field out. Unlike a PINN, it learns the solution operator for a family of PDEs and can be reused after one training run.

A neural operator that convolves in the Fourier domain: lift the input to frequency space, truncate to low-frequency modes, apply a learned linear transform, then invert back to physical space. Extremely efficient on regular grids and the most widely used neural operator in practice.

An operator-learning architecture with a branch net encoding the input function and a trunk net encoding the evaluation location; their inner product gives the solution value. It does not assume a regular grid and is the other mainstream neural operator design alongside FNO.

A neural operator that treats an unstructured mesh as a graph and approximates the integral kernel by message passing over neighborhoods. It targets complex geometries and adaptive meshes where FFT-based operators break down.

An acceleration technique that projects a high-dimensional simulation (millions of DOFs) onto a low-dimensional subspace and solves there. A handful of basis functions capture the dominant physical modes, cutting solve time from hours to seconds. One of the most mature AI-acceleration routes in industry.

A data-driven model that approximates the input-output behavior of an expensive simulator so that large numbers of evaluations become cheap. The emphasis is on "replacing" rather than "reducing": it usually does not exploit the governing-equation structure explicitly.

The umbrella term for machine learning that targets mappings between infinite-dimensional function spaces; neural operators are its dominant implementation. What separates it from ordinary supervised learning is that both inputs and outputs are functions, not vectors.

The squared residual obtained by substituting the network output into the PDE, used as a training loss term. It is the core mechanism of PINNs: it lets the network be constrained by the governing equation even with no labeled solution data.

A dimensionality-reduction method that extracts an energy-optimal orthogonal basis from a matrix of simulation snapshots: the fewest basis functions capturing the most field energy. It is the foundation of classical POD-Galerkin reduced-order models.

Training with a mix of cheap low-fidelity data and expensive high-fidelity data: a small number of high-fidelity samples corrects a large pool of low-fidelity samples, maximizing accuracy under a fixed budget.

Making an entire simulation pipeline — including the solver iterations — differentiable with respect to its inputs, so gradients can drive inversion, control and end-to-end training. JAX-CFD is a representative differentiable CFD framework.

Feeding a trained one-step model its own predictions repeatedly to advance many time steps. It is the standard way to do temporal field prediction and the main source of error accumulation.

Compress a high-dimensional field into a low-dimensional latent code with an encoder, learn the temporal evolution in latent space (ODE / neural ODE / RNN), then decode back to physical space. The dominant architecture for nonlinear reduced-order modeling.

The classical numerical method that discretizes a domain into finite elements and builds an approximate solution with shape-function interpolation inside each element. The de facto standard for solid mechanics and structural analysis, and the source of most Physics-AI training data.

A numerical method that integrates conservation laws over each control volume and builds discrete equations from flux balance. It guarantees conservation of mass, momentum and energy by construction, making it the dominant discretization for CFD and heat transfer.

The discipline and engineering practice of solving flow governing equations such as Navier-Stokes numerically, covering meshing, discretization, turbulence modeling and solution strategies. It is the primary target that Physics-AI aims to accelerate or replace in the fluid domain.

Numerical methods that build approximations directly on scattered points or particles without a pre-generated mesh. Suited to large deformation, free surfaces and fracture where meshes distort and fail, and a natural fit for PINNs and DeepONets.

Public datasets and metric suites for uniformly evaluating Physics-AI models, typically covering multiple PDE families, multiple parameter regimes and standardized train/test splits. A prerequisite for any credible claim that a model is better.

The discipline of characterizing the sources and magnitude of uncertainty in model predictions: separating input-parameter uncertainty (aleatoric) from model uncertainty (epistemic), and producing decision-usable confidence intervals rather than a single point prediction.

The ability of a model to remain usable outside its training distribution — more extreme parameters, longer horizons, unseen geometries. It is the biggest bottleneck taking Physics-AI from papers to engineering, and the weakness most easily hidden by evaluation metrics.

The umbrella term for the intersection of machine learning with scientific computing and physical modeling: both using AI to accelerate or replace simulation, and embedding physical knowledge into model structure. The organizing concept behind everything indexed on this site.

Inferring model parameters, source terms or boundary conditions from observations — the opposite of the forward problem of computing results from known parameters. Usually ill-posed (non-unique or unstable), yet among the most valuable problems in engineering.

A simulation model that maintains a two-way data connection with a physical asset, is continuously updated, and is used for prediction and decision-making. The emphasis is on being online and co-evolving with the asset, unlike one-off offline simulation or static surrogates.

Simulation that solves two or more mutually interacting physical fields (fluid-structure, thermal-mechanical, electro-magnetic-thermal) and exchanges coupling quantities across interfaces. Coupling convergence and multiscale time stepping are the main difficulties, and also where AI acceleration pays off most.

The phenomenon that neural networks fit low-frequency components first and struggle to learn high-frequency oscillations. It explains why PINNs converge so slowly on multiscale and high-frequency problems, and motivates adaptive weights, Fourier features and curriculum learning.