Coordinate-Free Automatic Differentiation

The gist Automatic differentiation (AD) is a cornerstone of modern machine learning, and this work introduces a single, basis-independent derivative operator that can represent many geometric differential operations and be expressed entirely in terms of existing AD primitives

How it works

The paper addresses the limitation of current AD systems which implicitly fix a basis of Euclidean space, making it awkward to express objects like rotations or curvature, and proposes extending this philosophy from flat-space AD to a fully coordinate-free formalism known as geometric algebra The core idea is to define a multivector derivative that incorporates all relevant metric and orientation information algebraically, which can then be implemented as a composable JAX transformation supporting forward and reverse mode AD, higher-order differentiation, and batched evaluation

The multivector derivative is formally defined as (2) ∂IF:= X J∈I b J ∂F/∂X J for a function F indexed by I, where the multiplication in the sum is the geometric product This operator is coordinate-free and intrinsic to Λ(V), unlike Euclidean Jacobians which live in R n 2

The derivative can be mapped to JAX primitives using Jacobian-vector products (jvp and vjp) Specifically, the metric aware vectorJacobian product is defined as (3) vjp⟨·,·⟩(F, X, C):= vjp(F, X, C = dF∗ X[C] where dF∗ X[C], A = C J, dFX[A] Lemma 2 shows that the multivector derivative arises from these primitives: ∂IF(X) = X J∈I b J jvp(F, X, bJ)

Geometric Operators and Manifolds

By specializing this derivative, the framework obtains automatic computation of various differential operators on any (pseudo-) Riemannian manifold in any number of dimensions These include curl, divergence, covariant derivatives, shape operators, and curvature tensors The approach generalizes from flat spaces to curved manifolds by representing a manifold M as a vector manifold where points are represented as vectors in Λ1(V) and the tangent space is given by a blade I(x) of grade m

The intrinsic derivative on the manifold is obtained by projecting the vector derivative operator ∂I1 onto the tangent space, resulting in (6) ∂M:= P(∂I1) where P(F)(x):= (F(x)I−1(x))I The covariant derivative is then defined as ∇MF:= P(∂MF) = P(P(∂I1)F) This ensures that standard differential operators on manifolds, such as the curl and divergence, are expressed in terms of ∂M and ∇M

Applications in Learning and Physics

The framework demonstrates reuse across different applications, including manifold learning, inverse problems in spacetime, and optimization over rotor groups In manifold learning experiments, a divergence-free vector field on a manifold can be constructed by starting with an arbitrary 2-form and applying ∇M⌋ to get a divergence-free vector field Furthermore, the framework allows for the estimation of geometric properties like scalar curvature R and Euler characteristic χM through compositions of ∂M and IM

In inverse design problems in spacetime, the spacetime algebra Cl(1, 3) is used to model electromagnetism where Maxwell's equations collapse into a single equation involving the multivector derivative This allows for recovering a charged particle’s trajectory r(t) from sparse Faraday bivector observations F(yi) by backpropagating the relative MSE between predicted and observed Faraday bivectors

Optimization on Rotor Groups

For optimization on rotor groups, the multivector derivative is used to compute the left-trivialized differential dfe(g) in terms of the multivector derivative as (10) dfe(g) = ∂Ieven f(g) g ∈ g∗ This allows for multiplicative descent where an update is performed via gt+1 = gt exp(−η dfe(gt)) The framework enables simultaneous optimization across different rotor groups by using JAX’s vectorization to optimize an objective across different groups in a single vectorized call

Overall, the main practical benefit of this approach is that geometric differential operators can be exposed as reusable AD transformations rather than being re-derived for each coordinate system or application This capability allows geometric structure to enter through the selected index set I, the multivector field being differentiated, and on manifolds, the tangent pseudoscalar IM

The paper concludes by showing that a small set of geometric operators can support several apparently different workflows across manifold learning, inverse problems in spacetime, and optimization on rotor groups The same implementation of the multivector derivative solves problems ranging from manifold learning to optimization on rotor groups

The final loss is a relative mean squared error between the predicted and observed F over the N observation events This allows for the recovery of a Lorentz transformation R ∈ Spin(1, 3)+ that maps the lab frame to the ship frame by minimizing a loss between analytic and measured fields The recovered worldline has a mean position error of 6 · 10−4 and a maximum of 1.2 · 10−3 over t ∈ [1, 4], i.e. 0.2% and 0.3% of the orbit radius The whole comparison, including the learning-rate sweeps, takes 20 s on an Apple M3 laptop CPU

Table 2 reports the results for the four subgroups of the conformal group of Fig. 9, fitted to one shared conformal target, noise-free (top) and with noise σ = 0.05 (bottom) Every on-group method reaches the noise floor, and on this task the three trivialized updates agree to all printed digits The penalized matrix also reaches the floor but leaves the group (constraint violation 1.9 · 10−3) The polar projection stalls at 48σ2 with violation 1.7, because O(4) contains no boosts Every method runs 400 Adam steps from the same initialization, with the learning rate chosen from a set of values as the smallest one within 5% of the best final loss The whole comparison, including the learning-rate sweeps, takes 20 s on an Apple M3 laptop CPU

The paper is published in arXiv as arXiv:2610.12776v1 [cs.

1. Improved AI systems can perform automatic computation of geometric differential operators on any (pseudo-) Riemannian manifold by leveraging a single, basis-independent derivative operator. This allows for automatic computation of curl, divergence, covariant derivatives, shape operators, curvature tensors and related operators on any (pseudo-) Riemannian manifold without needing to supply geometric structure manually.

2. AI systems can learn explicit representations of manifolds by minimizing losses involving geometric properties like scalar curvature and the Gauss–Bonnet theorem. This enables training a flow to match the sphere’s curvature results in learning a small section of the sphere, allowing for optimizing Eq. 8 to approach the Euler characteristic of the sphere.

3. AI systems can recover physical parameters from noisy geometric measurements by using multiplicative descent methods over rotor groups and spacetime algebras. This allows for recovering a Lorentz transformation R from Faraday bivector observations, with the framework demonstrating that updates converge to correct solutions even when starting far from the identity.

4. AI systems can learn implicit representations of shapes by parameterizing boundaries using k-forms and minimizing their pointwise mass norm. This enables learning surfaces where the boundary constraint is satisfied structurally via a GA-native Biot–Savart kernel, as demonstrated by successfully extracting minimal surfaces like the (2, 3) torus knot.

Automatic differentiation (AD) is a cornerstone of modern machine learning. Since all AD frameworks implicitly fix a basis of Euclidean space to define gradients and Jacobians, they natively process mathematical objects that are arrays of a fixed set of coordinates. This makes it awkward to express objects such as rotations, volume forms, and curvatures that arise, for example, in applications of AI to science. Bespoke workarounds lack reusability. We show that (i) a single, basis-independent derivative operator can represent many geometric differential operations, and (ii) this operator can be expressed entirely in terms of existing AD primitives. This reusable transformation can be composed with JAX batching, compilation, and hardware acceleration. Geometry enters as algebraic data. We experimentally demonstrate its reuse across different manifold operators, its composition through a spacetime inverse-problem pipeline, and its use in optimization over different rotor subgroups.