Automatic differentiation drops the coordinate basis: one JAX transformation now yields curvature, divergence and rotor-group updates

Automatic differentiation can now handle geometric objects such as rotations, volume forms and curvature directly, instead of first flattening them into fixed coordinate arrays. Existing automatic differentiation frameworks implicitly fix a basis of Euclidean space to define gradients and Jacobians, so they natively process arrays of a fixed set of coordinates; for fields on manifolds, rotations, curvature, volume forms, divergence, curl or covariant derivatives, the geometric structure had to be supplied by hand through a representation such as Christoffel symbols in a chart, cotangent weights on a mesh, or a stiffness matrix for a basis, and re-derived for each new space. The same multivector-derivative implementation is reused across different manifold operators, composed through a spacetime inverse-problem pipeline, and used for optimization over different rotor subgroups, showing the transformation carries across settings.

The construction expresses the multivector derivative in terms of existing automatic differentiation primitives, with geometry entering as algebraic data. The multivector derivative is defined as a sum of geometric products between basis blades and their reciprocals, then implemented with forward-mode jvp and reverse-mode vjp; reverse mode requires musical isomorphisms to turn the stored coordinate data into a metric-aware form. The implementation is built entirely from JAX primitives, so it composes with JAX batching, compilation and hardware acceleration; curvature, divergence and curl on manifolds follow from composing the projection operator with this derivative, with no coordinate charts or Christoffel symbols.

The demonstrations cover explicit and implicit manifold learning, a spacetime electromagnetic inverse problem, and optimization over rotor groups, with geometric structure entering through the selected index set, the multivector field being differentiated, and the tangent pseudoscalar on manifolds. It is aimed at researchers and engineers who need coordinate-free differential operators, especially those already working in JAX.

In high dimensions computational costs can become prohibitive, since the geometric-algebra space grows with dimension. No closed form is known for the mass norm beyond bivectors, so the implicit minimal-surface experiments only attempt the codimension-two case. Non-compact groups admit no bi-invariant metric, so the positive-definite metric choice is unavoidably non-invariant.