Physics-Informed Neural Networks glossary

Clear, one-line definitions of the Physics-Informed Neural Networks terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.

autograd

An automatic-differentiation engine (notably PyTorch's) that records the computational graph during the forward pass and replays it in reverse for gradients.

automatic differentiation

Exact, machine-precision evaluation of derivatives of a program by applying the chain rule to its elementary operations; the engine behind PINNs and deep learning.

boundary loss

PINN loss term that enforces boundary conditions, evaluated at points sampled on ∂Ω.

causality violation

When a numerical scheme or PINN allows information to propagate backwards in time or faster than the wave speed, breaking physical causality.

classical fwi recovery

The velocity model produced by standard full-waveform inversion using adjoint gradients, used as a benchmark against PINN-based inversion.

collocation points

Spatial-temporal sample points at which the PDE residual is evaluated during PINN training; sampled inside the domain Ω.

cycle skipping

Convergence failure of FWI in which the inversion locks onto the wrong wavelet cycle when the starting model lags the data by more than half a period.

See: Cycle skipping: detection and remedies

data loss

The component of a PINN or hybrid loss that measures misfit between predicted and observed quantities at sensor locations.

deeponet

An operator-learning architecture with a branch net (input function) and a trunk net (output location) whose dot-product approximates a non-linear operator G: u ↦ G(u).

See: DeepONet: branch and trunk networks

eikonal equation

|∇T(x)|² = 1/v(x)²; a first-order non-linear PDE governing first-arrival traveltimes T in a medium of velocity v.

See: The eikonal equation and why PINNs love it

fbpinn

Finite Basis PINN: a domain-decomposition variant that fits local PINNs on overlapping subdomains glued by partition-of-unity windows; scales to large or multiscale problems.

See: Domain decomposition: XPINN, cPINN, FBPINN

forward modelling

Solving the forward problem: given subsurface parameters (velocity, density), simulate the seismic data that would be observed.

See: Learned propagators for fast forward modelling

forward problem

Given physical parameters, predict observations by solving the governing PDE; the standard simulation direction.

See: Why Model? The Forward Problem, The forward problem vs the inverse problem

frequency continuation

PINN/inversion training strategy that injects low frequencies first and slowly enlarges the spectral band, analogous to multiscale FWI; mitigates spectral bias.

See: Multi-scale frequency continuation

gradient flow pathology

A PINN failure mode where gradients of one loss term vanish or explode through the network, stalling optimisation of that constraint.

gradient pathology

Failure mode of multi-term PINN losses where one term's gradient dominates another's; mitigated by adaptive weighting or learning-rate annealing.

hard constraint

Imposing a condition by constructing the network output so it satisfies the constraint exactly, e.g. u_θ(x, t) = g(x) + N(x, t)·u_NN(x, t).

helmholtz equation

(∇² + k²) u = 0; the frequency-domain wave equation with wavenumber k = ω/c, used in time-harmonic PINNs.

hyposvi

A Stein Variational Inference earthquake-location method that uses an eikonal-based neural traveltime surrogate to evaluate likelihoods quickly.

initial-condition loss

PINN loss term that enforces the initial condition u(x, 0) = u₀(x), evaluated at points sampled on the t = 0 slice of the domain.

inverse problem

Given observations, infer the physical parameters (velocity, density, source) consistent with them; often ill-posed and solved by optimisation.

See: The inverse problem, mathematically, The forward problem vs the inverse problem

loss-balance crisis

A PINN training pathology where data-misfit and physics-residual losses have wildly different scales, so one term dominates and the other is ignored.

See: The loss-balance crisis: data + PDE + BC weights

lr-anneal

Learning-rate annealing for PINN loss balancing (Wang et al.): adaptively rescales individual loss-term weights to equalise gradient magnitudes during training.

multiscale fwi

FWI strategy that inverts low frequencies first, then progressively higher ones, to mitigate cycle skipping by keeping data within half a period of the prediction.

parameter inversion

Estimating PDE parameters (e.g. seismic velocity) by jointly minimising data misfit and physics residuals over both network weights and the parameter field.

See: Multi-parameter inversion (vp, vs, rho)

pde

Partial Differential Equation: an equation relating an unknown function and its partial derivatives, e.g. the wave equation or Helmholtz equation.

physics-informed loss

The component of a PINN loss that penalises violation of a PDE, boundary condition, or other physical law at collocation points.

physics-informed neural network

A neural network trained with a loss that combines data fit and the residuals of a governing PDE evaluated by automatic differentiation at collocation points.

rad

Residual-based Adaptive Distribution: PINN sampling scheme that re-samples collocation points proportional to the residual magnitude rather than appending.

rar

Residual-based Adaptive Refinement: PINN sampling strategy that adds new collocation points where the PDE residual is largest, focusing capacity on hard regions.

residual loss

The mean-squared PDE residual term in a PINN, e.g. (1/N) Σ |𝒩[u_θ](xᵢ)|², ensuring the network output satisfies the equation in mean-square.

soft constraint

Imposing a condition (BC, IC, PDE) as a penalty term in the loss; the network may violate it slightly to reduce overall loss.

spectral bias

The empirical tendency of neural networks (especially MLPs with tanh) to learn low-frequency components first, struggling with high-frequency features in PDE solutions.

See: Spectral bias and why physics needs it fixed

traveltime

The arrival time T(x, x_s) of a seismic phase at receiver x from source x_s; the unknown in eikonal-based PINNs (e.g. EikoNet, HypoSVI).

truth velocity model

The reference subsurface velocity field used to generate synthetic seismic data in a controlled experiment; the target an inversion tries to recover.