Why bivector coordinate fields work

The continuum solver is Clifra's end-to-end research demonstration of layout-directed geometric learning. It implements a differentiable field of local Clifford actions parameterized by sampled bivector generators. Clifra supplies the structured coordinate spaces and compiled transformation law; user-defined differentiable objectives determine what the field learns.

The continuum construction extends a Clifford action into a generator field over a domain. Sampling determines how its local generators vary, paths determine how their actions compose, and differentiable objectives determine the transformation the field learns. The included physics-informed deformation system is one complete realization of this construction.

From a bivector to an action

Let \(B\) be a bivector in \(Cl(p,q,r)\), let \(n=p+q+r\), and hold \(B\) fixed over one action step. Define

\[ R(t)=\exp(-tB/2). \]

Because bivector reversal gives \(\widetilde{B}=-B\), the rotor action on a grade-1 value is

\[ x(t)=R(t)x(0)\widetilde{R(t)} =\exp(-tB/2)x(0)\exp(tB/2). \]

Differentiating gives

\[ \frac{d x(t)}{dt} =-\frac12\left(Bx(t)-x(t)B\right) =G(B)x(t). \]

Here, \(G(B)\) is the \(n\times n\) linear generator induced by the commutator on the grade-1 subspace. Therefore

\[ x(1)=\exp(G(B))x(0). \]

This is why bivector coefficients can be learned directly: the default planned action maps their linear coordinate space into invertible linear actions on grade 1. For an ordered path \(B_1,\ldots,B_S\),

\[ x_S=\exp(G(B_S))\cdots\exp(G(B_1))x_0. \]

The inverse uses \(-B_S,\ldots,-B_1\) in that order.

From one action to a field

CoordinateChart embeds input coordinates into grade 1 and extracts the result. Let \(\Theta\) denote the stored generator parameters. A sampler evaluates the step generator \(B_s^\Theta(\xi_i)\) at persistent sample label \(\xi_i\):

\[ \phi_\Theta(X_i,\xi_i) =D\!\left[ \exp\!\left(G(B_S^\Theta(\xi_i))\right)\cdots \exp\!\left(G(B_1^\Theta(\xi_i))\right)E(X_i) \right]. \]

CoordinateFieldInput keeps transformed values separate from persistent material labels. Regular-grid sampling associates generators by tensor index; RBF sampling associates them by arbitrary coordinates. Interpolation changes the generator field, never the input coordinate values.

Training is ordinary differentiation through the action:

\[ \min_\Theta\; \mathcal{L}\!\left(\operatorname{State}_\Theta(X,\xi),Y\right) +\sum_k\lambda_k \mathcal{P}_k\!\left(\operatorname{State}_\Theta(X,\xi),\Theta\right). \]

Here, \(\operatorname{State}_\Theta\) contains the input identity, transformed values, and sampled generator weights exposed by ContinuumState. A criterion or policy may use only the parts it needs. The mechanics example supplies one choice of \(\mathcal{L}\) and \(\mathcal{P}_k\). Dimension, signature, sampling domain, and objective are configurable parts of the construction.

Minimizing exponential work

The field's normal grade-1 action does not construct a full even multivector with bivector_exp. Clifra builds \(G(B)\) directly and evaluates the smaller torch.matrix_exp on the \(n\times n\) vector representation. The explicit rotor_path() and rotors_for_input() helpers use bivector_exp when a caller asks to inspect rotors.

The field also avoids exponentiating weights repeated only by broadcasting:

• a global generator is exponentiated once per path step;
• a regular-grid generator is exponentiated once per spatial site and shared across batch axes;
• a coordinate-dependent generator is exponentiated once per sampled generator row; numerically equal rows are not deduplicated.

Path steps are evaluated separately because, in general,

\[ \exp(G_2)\exp(G_1)\ne\exp(G_1+G_2) \]

when \([G_2,G_1]\ne0\). Merging such steps would change the parameterization. The implementation therefore preserves path composition and removes only broadcast-redundant exponential evaluations.

Local inversion and global geometry

For the default planned action, a path has an inverse when it is evaluated with the same persistent sample labels: negate its generators and reverse their order. Projective extraction additionally requires a nonzero homogeneous coordinate. An injected action module must satisfy the same inverse identity if it is to retain this property.

Global injectivity remains a property of the assembled field, not of each local matrix exponential. Application constraints may encourage it; the validation required by the application must establish the property actually claimed.

research/continuum_solver/examples/bivector_field_basics.py is the compact introduction; it learns a local field and verifies its structural properties. research/continuum_solver/examples/physics_informed_deformation_design.py is the complete physics-informed application.