finslerax — Differentiable Finsler Geometry in JAX

finslerax is a JAX-native library for learnable Finsler geometry. You supply a cost function $F(x, v)$ — the price of moving through point $x$ in direction $v$ — and the library derives what follows: geodesics, the geodesic spray, curvature, and parallel transport. Metrics are Equinox modules, so $F$ may be a neural network and the whole pipeline stays differentiable end to end.

Finsler geometry drops the Riemannian requirement that cost be symmetric. Travelling east can be cheaper than travelling west, which is what makes wind, ocean currents, directed representation dynamics, and anisotropic propagation expressible as geometry.

from finslerax.geometry import Randers, EuclideanSpace
from finslerax.solvers import AVBDSolver
import jax.numpy as jnp

# A plane with a steady eastward wind — moving with it is cheaper.
metric = Randers(EuclideanSpace(dim=2),
                 h_net=lambda x: jnp.eye(2),          # the "sea": flat Riemannian metric
                 w_net=lambda x: jnp.array([0.3, 0.0]))  # the "wind": a drift field

traj = AVBDSolver(iterations=50).solve(
    metric, jnp.array([0., 0.]), jnp.array([1., 1.]), n_steps=20)

print(metric.arc_length(traj.xs),          # downwind cost  ≈ 1.19
      metric.arc_length(traj.xs[::-1]))     # upwind cost    ≈ 1.85

Documentation: https://finslerax-docs.vercel.app/

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Features

• Define $F(x, v)$ and the geodesic spray, fundamental tensor $g_{ij}$, Berwald connection and flag curvature follow from jax.grad and jax.hessian. No hand-coded Christoffel symbols.
• The Euler–Lagrange equations are solved as a small linear system per step, avoiding the $O(N^3)$ cost of forming connection coefficients explicitly.
• Randers metrics carry a Zermelo parameterization that keeps the wind causal ($\lVert W \rVert_h < 1$) and the metric strongly convex, using a $C^\infty$ smooth-minimum that bends only near the boundary and leaves already-causal winds untouched.
• Four routes to a geodesic: shoot from initial conditions (ExponentialMap), relax a boundary-value path locally (AVBDSolver) or globally (GaussNewtonGeodesic), or solve the arrival-time PDE on a grid or mesh (the EikonalSolver family).
• Neural Riemannian and Randers metrics, decoder-pullback metrics for latent spaces, and energy-based and kernel wind fields are all trainable.
• Multi-phase training schedules with per-phase parameter freezing compose from a library of geometry-aware losses.

The HAM research programme

The library grew out of a programme called HAM (Holonomic Association Model): representing context as geometry. Parallel transport moves a representation between contexts along a path, and holonomy — the path dependence of that transport — is what the geometry remembers. Associations between states become geodesics of a learned, possibly asymmetric, metric. HAM remains the name of that research line; finslerax is the name of the software. The library is useful well beyond the programme, but the transport and holonomy machinery it needed is why every piece here exists.

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Prior art

The eikonal stack here follows Gahtan, Shpund & Bronstein, Differentiable Eikonal Wildfire Modelling (arXiv:2603.00035, code, MIT). Their algorithm and their reference implementation are what this was built from: fast sweeping forward, adjoint fixed-point backward, with the metric produced by a convolutional encoder over rasterized covariates. The JAX implementation and the Finsler generalization are new here; the design is theirs.

Three parts of this repository are directly downstream of that paper and say so in their docstrings:

• solvers/eikonal.py — the fast-sweeping anisotropic Godunov solver with jax.custom_vjp implicit gradients.
• models/covariate.py — LocalTerrainCNN follows their §6 encoder architecture; CovariateConditionedRanders uses their $\lambda = 1$ Zermelo parametrization.
• training/losses.py — the arrival-time metric-recovery loss follows their §5.

Foundational Finsler material follows Bao, Chern & Shen, An Introduction to Riemann–Finsler Geometry (Springer GTM 200, 2000); the anisotropic fast-marching literature of Jean-Marie Mirebeau informs the eikonal design.

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Installation

pip install finslerax

For development and the runnable examples:

git clone https://github.com/hubibala/finslerax.git
cd finslerax
pip install -e ".[dev]"          # core + dev tooling (pytest, ruff, matplotlib, …)
pip install -e ".[viz]"          # core + plotting (matplotlib, plotly) for finslerax.vis

The distribution is named finslerax; you import it as finslerax.

Extra	Installs	For
dev	pytest, ruff, mypy, matplotlib, jupyter, plotly	development and examples
viz	matplotlib, plotly	the finslerax.vis plotting helpers
gpu	jax[cuda12]	NVIDIA GPU acceleration

The core install carries only the geometry and solver stack: JAX, Equinox, Optax, NumPy, SciPy. JAX ≥ 0.4 is required; for GPU and TPU builds follow the JAX installation guide, of which the gpu extra covers the common CUDA 12 case.

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Numerical precision

Precision is governed by JAX's own jax_enable_x64 flag, which finslerax reads. The default is float32. To run in float64, set the standard environment variable before Python starts:

JAX_ENABLE_X64=1 python your_script.py

Equivalently, in-process before the first array is created:

import jax
jax.config.update("jax_enable_x64", True)
import finslerax   # now float64

That switch flips the whole stack, because precision is decided at the data-construction boundary and the solvers follow the dtype they are given. Stability floors (GRAD_EPS, PSD_EPS, TAYLOR_EPS, …) scale with it. Query the active setting through finslerax.utils.config:

from finslerax.utils.config import x64_enabled, default_dtype, default_np_dtype

Reach for float64 on stiff or ill-conditioned solves: long AVBD geodesics, fine eikonal grids, curvature and transport cancellation, tight VJP checks. Consumer NVIDIA GPUs throttle FP64 to roughly 1/32–1/64 of FP32 throughput, so float32 remains the default and float64 is opt-in.

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Quickstart

1. Shoot a geodesic on a sphere

Integrate the geodesic spray ODE from an initial position and velocity — the exponential map. Starting at the equator and shooting north for a quarter turn lands on the pole:

import jax.numpy as jnp
from finslerax.geometry import Sphere, Euclidean
from finslerax.solvers import ExponentialMap

sphere = Sphere(intrinsic_dim=2, radius=1.0)
metric = Euclidean(sphere)                  # round metric induced from the ambient norm

x0 = jnp.array([1.0, 0.0, 0.0])             # on the equator
v0 = jnp.array([0.0, 0.0, 1.0])             # unit velocity, pointing north

shooter = ExponentialMap(max_steps=200)
x_final = shooter.shoot(metric, x0, v0, t_max=jnp.pi / 2)
# x_final ≈ [0, 0, 1]  — the north pole (arc length |v0|·t_max = π/2)

A unit-speed geodesic travels t_max radians along a great circle, so reaching the pole from the equator needs t_max = π/2, not 1.0.

2. An asymmetric Randers metric

A Randers metric is a Riemannian "sea" $h$ plus a drifting "wind" $W$. Travel is cheaper downwind, so forward and backward arc length differ:

import jax.numpy as jnp
from finslerax.geometry import EuclideanSpace, Randers
from finslerax.solvers import AVBDSolver

manifold = EuclideanSpace(dim=2)
metric = Randers(manifold,
                 h_net=lambda x: jnp.eye(2),            # flat sea
                 w_net=lambda x: jnp.array([0.3, 0.0]))  # constant eastward wind

solver = AVBDSolver(iterations=50)
traj = solver.solve(metric, jnp.array([0., 0.]), jnp.array([1., 1.]), n_steps=20)

L_fwd = float(metric.arc_length(traj.xs))
L_bwd = float(metric.arc_length(traj.xs[::-1]))
print(f"downwind: {L_fwd:.4f}, upwind: {L_bwd:.4f}")
# downwind: 1.1890, upwind: 1.8484  — wind helps eastward travel

3. Learn a metric from data

A neural metric is an eqx.Module, so it trains in a standard Equinox and Optax loop. Minimizing the Finsler energy of observed (position, velocity) pairs makes the directions the data actually moves in cheap, which recovers the underlying drift:

import jax, jax.numpy as jnp, optax, equinox as eqx
from finslerax.geometry import EuclideanSpace
from finslerax.models.learned import NeuralRanders

key = jax.random.PRNGKey(42)
metric = NeuralRanders(EuclideanSpace(dim=8), key, hidden_dim=64, depth=3)

opt = optax.adam(1e-3)
opt_state = opt.init(eqx.filter(metric, eqx.is_array))

@eqx.filter_jit
def step(m, X, V, state):                          # X: (B, 8) points, V: (B, 8) velocities
    def loss_fn(m):
        return jnp.mean(jax.vmap(m.energy)(X, V))   # make observed motion low-cost
    loss, grads = eqx.filter_value_and_grad(loss_fn)(m)
    updates, state = opt.update(grads, state, m)
    return eqx.apply_updates(m, updates), state, loss

examples/demo_learned_wind.py has a full runnable version that recovers a Rossby–Haurwitz wind on the sphere with smoothness regularization.

For generative latent-geometry models — a VAE whose latent space carries a learned Randers metric — finslerax.training.TrainingPipeline orchestrates multi-phase training with per-phase freezing and geometry-aware losses such as ZermeloAlignmentLoss and EulerLagrangeResidualLoss. Those losses expect a model exposing encode, decode and metric rather than a bare metric; see spec/ARCH_SPEC.md § 6.

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Core concepts

finslerax separates where you are (topology) from how costly motion is (geometry), then layers solvers on top.

Layer	Abstraction	Concrete types
Topology	Manifold	EuclideanSpace, Sphere, Torus, Hyperboloid, Paraboloid, TriangularMesh
Geometry	FinslerMetric → AsymmetricMetric	Euclidean, Riemannian, Randers, DiscreteRanders, SegmentQuadratureMetric, ProjectivelyFlatRanders
Learnable geometry	subclasses of the above	NeuralRanders, NeuralRiemannian, PullbackRanders, PullbackRiemannian, KernelWindField, …
Geodesics	initial- and boundary-value solvers	ExponentialMap, AVBDSolver, GaussNewtonGeodesic, GeodesicLearningSolver
Arrival times	anisotropic eikonal PDE	EikonalSolver (grid), MeshEikonalSolver, VolumetricEikonalSolver (3D)
Transport and curvature	derived geometry	BerwaldConnection, curvature_tensor, riemannian_curvature, ricci_curvature, flag_curvature, sectional_curvature

Every FinslerMetric is an eqx.Module, hence a JAX PyTree, so any metric — including a neural one — passes straight through jax.jit, jax.grad and jax.vmap.

A geodesic problem comes in two forms. Shooting takes a start point and velocity and integrates the spray ODE, via ExponentialMap. Connecting takes two endpoints and finds the minimizing path, via AVBDSolver (local block descent), GaussNewtonGeodesic (global and second-order, with an iteration count independent of path length), or the eikonal solvers, which return a full arrival-time field by fast sweeping.

The mathematics is in spec/MATH_SPEC.md; the software design is in spec/ARCH_SPEC.md.

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Repository structure

src/finslerax/
├── geometry/     # Manifold and FinslerMetric ABCs, manifolds/, mesh, zoo/,
│                 # transport (Berwald connection), curvature (tensor/Ricci/flag/sectional)
├── models/       # learned.py: neural, pullback, energy-based, kernel metrics
│                 # covariate.py: CovariateConditionedRanders, terrain CNN
├── nn/           # VectorField, PSDMatrixField, RandomFourierFeatures, EBM, KDE
├── solvers/      # geodesic (IVP), avbd + gauss_newton + geodesic_learning (BVP),
│                 # eikonal / mesh_eikonal / volumetric_eikonal (arrival times),
│                 # continuation, graph_init, coloring (warm-starts)
├── training/     # TrainingPipeline, TrainingPhase, geometry-aware losses
└── sim/ utils/ vis/   # analytic fields, numerics, terrain, plotting

examples/        # runnable demo scripts + Jupyter notebooks
spec/            # MATH_SPEC.md, ARCH_SPEC.md
tests/           # 349 tests across 35 modules, run in both precisions

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Examples

Runnable scripts live in examples/; narrated walkthroughs with plots live in examples/notebooks/.

Topic	Script	Notebook
Geodesic shooting on curved surfaces	—	demo_geodesic_shooting.ipynb, demo_curved_manifolds.ipynb
Zermelo navigation / Randers winds	demo_zermelo.py	demo_zermelo.ipynb
Vortex wind field	demo_vortex.py	demo_vortex.ipynb
Learned wind from data	demo_learned_wind.py	demo_learned_wind.ipynb
Discrete (mesh) Zermelo metric	demo_discrete_zermelo.py	demo_discrete_zermelo.ipynb
Anisotropic eikonal fronts	demo_eikonal_fronts.py	demo_eikonal_fronts.ipynb
Parallel transport and holonomy	—	demo_parallel_transport.ipynb
Curvature: flag, Ricci, sectional	—	demo_curvature.ipynb
High-dimensional latent geodesics	—	demo_high_dim_latent_geodesics.ipynb
Generic neural Finsler metric	—	demo_generic_finsler.ipynb

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Tests

python -m pytest tests/ -q                       # float32 (default)
JAX_ENABLE_X64=1 python -m pytest tests/ -q      # float64

Every test passes under both precisions. Precision-sensitive tolerances and dtype checks adapt through tests/_precision.py (tol(), assert_default_dtype), and the active precision is printed in the pytest header. CI runs the full matrix: JAX_ENABLE_X64 ∈ {0, 1} × Python 3.10/3.11, on CPU. If you hit accelerator initialization in a CPU-only environment, set JAX_PLATFORMS=cpu.

Module	Covers
test_metric.py, test_zoo.py	metric algebra, spray, energy
test_geodesic.py, test_solver.py	spray ODE, energy conservation
test_avbd.py, test_gauss_newton.py	BVP solvers, implicit differentiation
test_eikonal_solver.py, test_mesh_eikonal.py, test_volumetric_eikonal.py	arrival-time PDEs
test_transport.py, test_curvature.py	parallel translation, curvature
test_pipeline.py, test_learned_metric.py	training pipeline, neural metrics
test_invariants.py	Finsler axioms and cross-cutting metric invariants

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Contributing

Contributions are welcome. See CONTRIBUTING.md for the development workflow and contribution policy.

Citation

@software{finslerax2026,
  author = {Hubicska, Bal\'azs},
  title  = {finslerax: Differentiable Finsler Geometry in JAX},
  year   = {2026},
  url    = {https://github.com/hubibala/finslerax}
}

License

MIT — see LICENSE.