turing-surface#

Reaction–diffusion systems (Turing patterns) on closed surfaces given by spherical-harmonic embeddings, solved live in the browser with a spectral method whose transforms run on the GPU via WebGPU.

This is the sibling of turing-sphere, which solves the same systems on the round sphere. Everything there is here; what is added is a surface.

Warning

The geometry is rendered, not yet solved on. The Laplace–Beltrami operator in the models is still the round sphere's — the term that carries the shape is a placeholder that is identically zero. On anything but the sphere you are looking at the sphere's pattern painted onto that surface, not the pattern that surface would grow. Everything the correction needs in order to be dropped in — the embedding, the split of the operator, the iterative solve, the unrolled loop — is built and tested. See The geometry is not in the operator yet.

What a surface is here#

A geometry is an embedding of the sphere into R³: three scalar fields x, y, z over the (θ, φ) parametrization, each carried as spherical-harmonic coefficients. The unit sphere is the case where all three are pure degree-1 harmonics.

You write one down as MATLAB, in geometries/:

function [gx, gy, gz] = shape(theta, phi, waist, stretch)
  st = sin(theta);
  r = 1 - waist * (st .^ 2);
  gx = r .* (st .* cos(phi));
  gy = r .* (st .* sin(phi));
  gz = (1 + stretch) * (r .* cos(theta));
end

That is ordinary element-wise MATLAB and goes through the same compiler and the same WGSL backend the models do. It is evaluated once on the solver's grid, and then analysed into coefficients, which is the form everything downstream uses. Two things follow from going through the coefficients rather than keeping the pointwise values:

• It is exactly band-limited at lmax. The surface has as many derivatives as the scheme needs and no aliased content the solver cannot see. What the solver and the renderer both use is the synthesis of the coefficients, so for a shape with sharp features the surface being solved on is not quite the one that was written down — which is the honest thing for a spectral method to do.
• It can be evaluated on any grid. The renderer draws the surface on the (possibly finer) display grid by synthesizing the same coefficients there. That is exact interpolation, not subdivision — the same argument that lets the species fields be oversampled, and it is checked directly in the tests.

Four geometries ship: sphere (the reference case), ellipsoid, peanut — a dumbbell whose waist is a saddle — and bumpy. Each is editable in the page, with its own parameters. Changing a shape does not recompile the solver and does not disturb the run: the geometry is data whose shape in the bindings depends only on the grid, so a swap is six buffer writes and the pattern carries straight on.

A morph slider blends the drawn surface back to the unit sphere. The parametrization is the sphere's either way, so sweeping it shows which point went where.

The scheme, and where the geometry enters#

It solves the N-species system

d(u_k)/dt = D_k*lap_g(u_k) + f_k(t, u_1, ..., u_N),    k = 1, ..., N

where lap_g is the Laplace–Beltrami operator of the surface. On the round sphere lap_g is diagonal in spherical-harmonic space with eigenvalues -l(l+1), which is what makes turing-sphere's implicit diffusion a single divide. On a general surface it is not diagonal, and not even constant- coefficient, so that divide has to become a solve.

The models split the operator:

lap_g = lap_s + dlap

with lap_s the round-sphere one. (I - dt*D*lap_s) is still exactly invertible, so the implicit step

(I - dt*D*lap_g) Unew = B

rearranges into a fixed point that keeps the whole geometry on the right-hand side,

Unew = (B + dt*D*dlap(Unew)) ./ (1 + dt*D*lam)

and the loop iterates it from the round-sphere answer. That is preconditioned Richardson, with the operator we can invert exactly as the preconditioner; it converges while dt*D*dlap stays small against (I - dt*D*lap_s), which is what would keep the cost to a few transforms per step rather than a full elliptic solve. Written out, the whole of models/schnakenberg.m's step is:

function [Un, Vn, u, v] = step(U, V, lam, gx, gy, gz, a, b, D1, D2, dt, niter)
  u = synth(U);
  v = synth(V);
  uuv = u .* u .* v;

  Bu = U + dt * analys(a - u + uuv);
  Bv = V + dt * analys(b - uuv);

  Un = Bu ./ (1 + (dt * D1) * lam);
  Vn = Bv ./ (1 + (dt * D2) * lam);

  for k = 1:niter
    dLu = 0 * Un;                                    % <- the placeholder
    dLv = 0 * Vn;
    Un = (Bu + (dt * D1) * dLu) ./ (1 + (dt * D1) * lam);
    Vn = (Bv + (dt * D2) * dLv) ./ (1 + (dt * D2) * lam);
  end
end

Written this way rather than as a residual correction on purpose: with dlap zero, every iterate is bit for bit the first line, with no cancellation to round differently. So the sphere case is not "close to" turing-sphere, it is the same arithmetic, and the tests assert exactly that — the state after 20 steps is identical at 0, 1 and 4 iterations.

The geometry is not in the operator yet#

What belongs where dLu is now is dlap = lap_g - lap_s applied to the current iterate. Getting it needs two things this repo does not have:

1. The induced metric, g_ij = ∂_i X · ∂_j X for X = (gx, gy, gz). The geometry is static and low-degree, so this is a one-off precomputation, not per-step work — but it needs θ- and φ-derivatives of the embedding.
2. Surface derivatives of the field, per iteration. In the round frame this is the spheroidal transform pair — SHTNS's SHsph_to_spat and spat_to_SHsph, i.e. grad_s and div_s — which lets the operator be written as div_s(A grad_s f) with A built from the metric, with no explicit 1/sin θ to go singular at the poles.

Both need Legendre derivative tables, which the vendored WGSL transforms under src/sht/ do not implement — they are scalar synthesis and analysis only. That is the missing piece, and it is a substantial addition to the transforms rather than a change to the models. Until it lands, the models take gx, gy, gz (the surface on the grid) and Gx, Gy, Gz (the same surface as coefficients) as arguments and do not use them, and the app says so.

for loops, unrolled#

A plan is a fixed list of GPU operations with no branching, which is what makes a timestep pure command recording — one submit, no CPU in the loop. A counted loop still fits: the planner (src/mgpu/plan.ts) unrolls it, planning the body once per iteration.

Nothing else had to change for that, because numbl gives a variable one cName for every assignment to it: the buffer an iteration writes is the buffer the next one reads, which is exactly a loop-carried value. The loop variable gets no buffer at all — it is bound as a derived scalar to that iteration's literal, so a kernel reading k folds the number in.

Two consequences worth stating:

• The bounds must be known when the model compiles. niter is supplied as a fixed scalar rather than a tunable one, so changing it recompiles — unlike a parameter, which is a uniform. A runtime bound is refused at compile time with a source position, not silently mis-compiled, and there is a test for that.
• Fusion survives. numbl's inline pass recurses into loop bodies, so a line inside the loop is still one kernel. It runs there with no protected names, though, which means an assignment whose only visible use is later in the same body can be elided — correct for a body-local temp, wrong if something outside the loop wanted it. src/mgpu/compile.ts snapshots what each loop body assigns before the pass and refuses the ones that escape, so that case is a compile error rather than a stale read.

Unrolling is exactly linear in the trip count: 2 GPU ops per species per iteration, asserted in the tests.

MATLAB, compiled to WebGPU#

Unchanged from turing-sphere, and it now compiles the geometry files too. numbl parses and lowers each function for the concrete argument types of the current grid; its inline pass folds single-use temps back into their consumer, so one line of MATLAB becomes one expression tree; and this repo emits one WGSL compute kernel per element-wise statement (src/mgpu/wgsl.ts). synth / analys are external operations whose type rules numbl learns from a .mtoc2.js workspace file, and which the backend maps onto the spherical-harmonic pipelines. Anything it cannot express is refused at compile time with a source position.

The Schnakenberg step above compiles to 17 GPU operations at one solve iteration: 4 transforms, 11 generated kernels, and 2 buffer copies feeding the new state back.

Two consequences carried over:

• The step is synchronous. WebGPU's encode path is synchronous and every pipeline is built once at compile time, so a timestep is pure command recording; the only await in the loop is the single readback per rendered frame.
• Parameters are uniforms, not constants. Moving a slider rewrites a small buffer instead of triggering a recompile. Editing the MATLAB recompiles; changing dt does not. niter is the deliberate exception, above.

Provenance#

• turing-sphere, which this is a fork of: the solver, the transforms backend, the compilation path, the benchmarks and the analytic tests.
• Transforms: shtns-webgpu — fp32 spherical harmonic transforms in WGSL compute shaders, modeled on SHTNS. Vendored under src/sht/ (CECILL-2.1), including the f64 CPU reference transform used for testing.
• Rendering: three.js meshes with per-vertex colormaps, adapted from the SphereEmbedding view in figpack's experimental extension package (src/render/). That view displays a time-varying embedded geometry with fields on it, which is the same picture this draws — including its sphere/surface morph, which turing-sphere had dropped as having nothing to morph to.

turing-sphere additionally carries a comparison against a native build of upstream SHTNS (bench/shtns/). That is not duplicated here: the transforms are the same code, and its C-side transcription of the model would have to be maintained against a step this project intends to change.

Because the algorithm is compiled to compute shaders, WebGPU is required — there is no CPU fallback (the f64 CPU transform remains, for tests).

Numerics#

• Grid: Gauss–Legendre × equispaced-φ, dealiased for the cubic reactions with the (pdeg+1) rule: nlat ≥ ((pdeg+1)·lmax+1)/2, nphi ≥ (pdeg+1)·lmax+1 (rounded up to a power of two for the GPU FFT path). At the default lmax 63 that is a 128×256 grid.
• Spectral layout: SHTNS conventions — orthonormal + Condon–Shortley, complex coefficients for m ≥ 0, m-major ordering.
• fp32 transforms introduce ~1e-6 relative error per step; for pattern formation from 1e-2 seeded noise this is inconsequential. The geometry goes through one analysis/synthesis round trip and picks up the same round-off: the unit sphere comes back with radius 1 to ~2e-5 under Dawn, ~4e-4 under SwiftShader.
• The shipped geometries are all degree ≤ 5, far below any lmax the app offers, so band-limiting removes nothing from them. A shape you write yourself may not be so lucky — see the note in geometries/bumpy.m.

Desktop vs browser#

scripts/bench.ts runs the same thing the app runs — same .m, same generated WGSL, same transforms — from Node on desktop WebGPU (Google Dawn), and the app prints the command line that reproduces whatever it is currently simulating:

npm run bench -- --preset schnak-spots --geometry ellipsoid --lmax 63 --niter 1 \
  --steps 2000 --seed 1 --a 0.1 --b 0.9 --D1 0.0004 --D2 0.008 --dt 0.05 \
  --gax 1.5 --gay 1 --gaz 0.6

Copy it from under the stats line and compare the ms/step it reports with the app's. Both sides go through the one shared src/bench/runSpec.ts — the app formats a run into that command, the benchmark parses it back — so there is no second copy of the defaults for the two runs to drift apart on. Geometry parameters take a g prefix (--gwaist) so a shape parameter can never collide with a model one.

The app reports two numbers and only the first is comparable to the benchmark: solver is the batch of steps alone, waited for but not read back; ms/frame additionally carries a GPU→CPU readback per species, the colormapping, and the vertex upload. Those per-frame costs are fixed and do not shrink when the GPU gets faster, so on a quick GPU a frame can easily cost ten times the steps inside it. That is expected and is not the solver being slower in the browser.

To attribute the gap rather than guess at it:

node scripts/compare-perf.mjs [--lmax 63] [--steps 300]

measures the same solver work in both — batched, nothing read back, no rendering on either side — and reports each with its CPU-encoding share, the Fourier stage, and the adapter. It stops you first if the two are not even the same device, which is a common cause of "the browser is much slower". Both sides resolve the geometry and the iteration count from the same constants, because the iteration count is unrolled into the step and a mismatch would compare two different amounts of work.

The app's Benchmark button runs the same measurement in the page, plus the ramp — the first third of the run against the last. GPUs downclock when idle and an animation-paced loop leaves them idle most of every frame, so a large ramp means the steady-state number is limited by clocks rather than work.

Is it really the same computation?#

node scripts/compare-env.mjs [--lmax 31] [--steps 200] [--preset schnak-spots]

runs one identical spec on the desktop and in a real browser and compares the final spectral state. The pipeline is deterministic given (model source, geometry, parameters, lmax, niter, seed, steps), so the two should agree to fp32 round-off — not bit for bit, since GPUs differ in fused-multiply-add and other latitude fp32 allows. It also reports which Fourier stage each side chose, since FFT and DFT are genuinely different algorithms that round differently.

Desktop WebGPU comes from the webgpu package (prebuilt Dawn, ~70 MB), an optional dependency so that an unsupported platform fails the install of that package alone. Its binaries need glibc 2.29+. Other flags: --steps, --warmup, --batch, --json, --help; DAWN_FLAGS='backend=vulkan' (;-separated) passes Dawn options through.

Tests#

There is no second implementation of the solver to diff against, so the .m path is checked against closed-form answers and against exact structural properties. Four modules, run in both environments:

test/analyticChecks.ts — cases whose evolution is known exactly, run through the whole real pipeline. All three are statements about the round sphere, so all three build on the sphere geometry:

• A — a linear reaction leaves every mode independent, growing by exactly (1 + dt*c) / (1 + dt*D*l(l+1)) per step. Pins the transform round trip, the eigenvalue mapping, the IMEX update and the state feedback at once. ~2e-7 over 20 steps.
• B — a nonlinear reaction on a uniform field stays uniform, so each step is exactly the scalar ODE map. 1.5e-8 over 25 steps.
• C — a 1e-6 perturbation of the Schnakenberg fixed point follows the linearized 2×2 IMEX recurrence, and (l=24, m=7) is confirmed unstable. Looser (~4e-3) because fp32 keeps about four digits of a perturbation that small.

test/geometryChecks.ts — the surface and the loop:

• every geometry compiles and closes; the sphere has radius 1 everywhere and is exactly degree 1 in the harmonics, which is what makes the reference case exact rather than merely accurate;
• the peanut matches its own closed-form radial profile at every grid point, and the same coefficients give the same surface on a 2× grid — the 2× Gauss latitudes share no point with the 1× ones, so agreeing there is agreeing everywhere, which is what "rendered exactly, not subdivided" means;
• unrolling is exactly linear in the trip count, and the state after 20 steps is bit-identical at 0, 1 and 4 iterations;
• a runtime loop bound is refused at compile time;
• swapping the surface mid-run leaves the spectral state untouched.

test/modelChecks.ts compiles every model the app offers and asserts how many kernels it compiles to, split into the base step and what one solve iteration adds. That is a fusion guard: if numbl's inline pass stops folding, the results stay correct while every operator becomes its own dispatch, which is invisible in the numbers.

test/transformChecks.ts compares the WGSL transforms against shtns-webgpu's f64 CPU twin.

• npm run test:node — under Dawn on the desktop, via vite-node. Needs a GPU; --skip-without-gpu lets a machine without one say so and move on (which is what CI does, since the browser suite covers the same modules).
• npm run test:gpu — builds and drives headless Chrome, on SwiftShader in CI. Also runs the soak. A few geometry tolerances are set by SwiftShader's fp32, which is about an order of magnitude looser than Dawn's.

Other commands:

• npm run bench -- --help — the desktop benchmark.
• npm run bench:sht -- --help — the transforms alone, no solver.
• npx vite-node scripts/diagnose-sht.ts — when the transform tests fail on a GPU, say which stage is wrong.
• npx vite-node scripts/diagnose-leg.ts [--m 0] — read the Legendre recurrence out of the production shader term by term.
• npx vite-node scripts/longrun-node.ts [lmax] — run to t = 100 and confirm the pattern saturates rather than decaying or diverging.
• node scripts/soak.mjs [steps] [lmax] — drive the demo for many steps, sampling JS heap and catching crashes.
• node scripts/screenshot.mjs out.png [light|dark] [minSteps] — screenshot the demo after a number of steps.
• node scripts/check-live.mjs [url] — smoke-check a deployed URL.
• test.html?soak=<steps>&lmax=<n> — solver-only soak with no rendering.

Testing against a reference implementation#

Reference solutions live in the sibling turing-surface-test-data repo, so an independently-written solver never has to depend on this one. cases/schnakenberg-ellipsoid.md there specifies the one case this repo currently ships a reference for; docs/ellipsoid-reference-spec.md restates it in this repo's own terms.

npm run ref -- --in <file> (scripts/ref.ts) loads a reference file, runs the solver from its exact initial spectral state to the same physical end time, and reports the relative-L2 error against its final state (plus a geometry sanity check). --niter overrides the surface-correction iteration count independent of the file, and --tolerance turns the check into a pass/fail for CI.

Development#

npm install
npm run dev       # local dev server
npm run build     # type-check + production build to dist/

The numbl dependency#

numbl is a local file:../../numbl dependency, so a sibling checkout of numbl is required. We use its compiler internals — parser, lowerer, IR, inline pass — which its package exports map does not publish, so they are reached through the numbl-src path alias in vite.config.ts.

The exact surface we depend on is written down in src/mgpu/numbl.d.ts and TypeScript checks against that, not against numbl's sources. This keeps this project's compiler settings independent of numbl's, and means a change to one of those shapes upstream breaks the build here with a clear diff rather than deep inside numbl's tree. The For IR node is spelled out there, since the planner now walks it.

CI clones numbl to the sibling path that the file: dependency expects, pinned to a commit, with --ignore-scripts (npm runs a linked package's prepare script, and numbl's is husky). numbl's own node_modules are not needed: the slice we import is self-contained TypeScript.

The scripts/*.ts entry points that touch the compiler go through vite-node, so they resolve imports exactly as the browser build does. Plain node cannot: numbl's sources import each other as ./foo.js while the files are .ts.

Deployed to GitHub Pages by .github/workflows/deploy.yml on push to main.

License#

CECILL-2.1 (inherited from SHTNS via shtns-webgpu, whose sources are vendored).