Fields and domains

Use this page when an array is described as a physical field and you need to state the domain, value space, units, components, and boundary conditions that give the array scientific meaning.

The scientific question¶

What quantity is attached to each point of which domain, and how is that continuous or discrete object represented in a computation? A temperature field on a volume, a velocity field on mesh faces, and a spectral intensity field on directions and wavelengths may all be arrays, but they have different domains, components, units, and boundary semantics.

A sampled array is not by itself a field contract. Shape does not say whether an axis indexes coordinates, vector components, cells, faces, time, or independent systems. Values cannot be interpreted or differentiated responsibly until those roles and their conventions are explicit.

Mathematical objects¶

Let $\Omega$ be a domain and $V$ a value space. A scalar field has one value per domain point; a vector or tensor field has components defined relative to a basis or frame. Domain points are geometric objects. Coordinates are labels assigned by a chart and can change without changing the underlying point. Field values may carry physical units distinct from coordinate units.

A computational representation also needs a sampling location: nodes, cell centers, faces, or edges. It needs component ordering and basis conventions, including whether vector components are global Cartesian components or local basis components. It needs boundary conditions, such as periodic, Dirichlet, Neumann, inflow, outflow, reflective, or domain-specific constraints. A boundary condition is part of the operator problem, not padding metadata.

The domain may be continuous while its discretization is finite. Conversely, a graph or catalog can be a fundamentally discrete domain. The contract must not invent a continuous geometry when only adjacency is meaningful.

Core derivation¶

The defining object is a map from domain points to values:

Equation (1) separates three things that array notation often collapses: the point $p\in\Omega$, its coordinate representation in a chosen chart, and the value $\phi(p)\in V$. If coordinates change from $x$ to $x'$, a scalar value at the same point is unchanged, while vector and tensor components transform according to their basis convention.

Sampling at points $p_i$ produces values $\phi_i=\phi(p_i)$, but the finite list $\{\phi_i\}$ does not reconstruct the field without the points, sampling locations, interpolation or basis rule, domain boundary, and component convention. Even on a regular grid, transposing axes while retaining shape can change the physical map.

Units provide a useful audit. If $p$ uses length coordinates and $\phi$ is density, then $\phi$ has mass per volume regardless of array dtype. Integrating it requires a measure or cell volumes with volume units; summing samples alone does not produce a mass.

Failure modes and interpretation limits¶

• Treating array indices as physical coordinates silently assumes origin, spacing, ordering, and chart.

• Omitting component basis information makes vector or tensor values ambiguous.

• Mixing node-centered and cell-centered values can introduce half-cell shifts and invalid operator stencils.

• Ignoring boundary conditions produces under-specified gradients, fluxes, and evolution problems.

• Interpolating across coordinate singularities or domain cuts can violate the intended geometry.

• Reusing one array for values with incompatible units makes reductions and losses scientifically meaningless.

• Assuming that a finite sample uniquely identifies a continuous field hides the reconstruction model and its approximation error.

What Jaxstro may add¶

Current jaxstro.spatial, grid utilities, mesh utilities, geometry, and operator utilities remain narrower owners of their existing contracts. They do not compose into a general field abstraction merely because each works with arrays in space. jaxstro.fields does not exist, and these pages do not propose one as imminent.

Only after two real consumers demonstrate the same domain, topology, value-location, component, unit, boundary, and transform needs could Jaxstro consider a common domain-agnostic representation. Such a representation might separate static domain metadata from dynamic field values and make sampling location explicit. It must not absorb solver policy, domain equations, or downstream physical interpretation.

Evidence required before implementation¶

Two independent consumers must first supply concrete use cases and show that one contract serves both without optional-field sprawl. Evidence must include scalar, vector, and where justified tensor values; coordinate changes; unit checks; node-, cell-, and face-located data; and multiple boundary types. Round-trip tests must show that serialization preserves domain identity, coordinates, component basis, sampling location, units, and boundary metadata.

JAX evidence must distinguish static topology and convention data from dynamic value leaves under JIT, VMAP, and differentiation. Analytic fields should verify sampling and coordinate transformation. Failure tests must reject axis, unit, location, and boundary mismatches before an operator returns plausible numbers.

Claim boundary¶

This page defines a field concept, not a package, base class, mesh standard, solver, or schedule. It does not claim that existing arrays are fields or that all domains share one useful runtime representation. The two-consumer rule is an evidence gate: shared abstractions follow demonstrated common structure rather than anticipated reuse.

Connected representations, foundations, and methods¶

• Return to Scientific representations for current representation families and their runtime owners.

• Review Functions, units, and scales for maps, domains, codomains, and units.

• Compare Vector geometry and rigid transforms for current geometric conventions.

• Continue to Topology and discretization and Grids and conservative bin transfer to separate fields from the discrete structures used to store them.