2dedc35turing-sphere: reaction-diffusion on the sphere, spectral solver on WebGPUJeremy Magland 1# turing-sphere
3Reaction–diffusion systems (Turing patterns) solved **live in the browser on the
4surface of a sphere**, using a spectral spherical-harmonic method with the
5transforms running on the GPU via WebGPU.
61e12f1Write the solver in MATLAB and compile it to WebGPUJeremy Magland 7The solver itself is **MATLAB**. The `.m` files under [`models/`](models/) are the
8algorithm — [numbl](https://numbl.org) parses and lowers them in the browser, and
9each element-wise line becomes a WebGPU compute kernel. You can edit the MATLAB
10on the page and watch the pattern change.
c6376dePoint Pages at the new repo and stop asking the workflow to enable itJeremy Magland 12**Live demo:** <https://concept-collection.github.io/turing-sphere-2/>
14## What it does
16It solves the N-species system
18```
19d(u_k)/dt = D_k*lap_s(u_k) + f_k(t, x, y, z, u_1, ..., u_N), k = 1, ..., N
20```
22on the unit sphere, where `lap_s` is the Laplace–Beltrami operator. Diffusion is
23treated implicitly in spherical-harmonic coefficient space, where `lap_s` is
24diagonal with eigenvalues `-l(l+1)`; reaction is treated explicitly on the grid.
25The two are combined with a first-order IMEX Euler step — the entire time loop is
27```
28V_k = synth(U_k) # spectral -> grid
29R_k = analys(f_k(t, x, y, z, V_1..V_N)) # reaction on grid -> spectral
30U_k = (U_k + dt*R_k) / (1 + dt*D_k*l(l+1))
31```
33You watch the patterns emerge in real time on orbitable 3D spheres (one per
34species, cameras synced), with pause/resume, re-seeding, live parameter editing,
35and colormap selection.
61e12f1Write the solver in MATLAB and compile it to WebGPUJeremy Magland 37Three models are included, one `.m` file each:
39- **[Schnakenberg](models/schnakenberg.m)** — Turing spots (unstable band
40 14 ≤ l ≤ 40, peak l = 24)
41- **[Brusselator](models/brusselator.m)** — stripes and spots from a stiffer reaction
42- **[Allen–Cahn](models/allencahn.m)** — a single species whose interfaces form
43 and coarsen
45## MATLAB, compiled to WebGPU
61e12f1Write the solver in MATLAB and compile it to WebGPUJeremy Magland 47A model file is ordinary MATLAB defining two functions — `init` builds the initial
48spectral state, `step` advances it one timestep:
50```matlab
51function [Un, Vn, u, v] = step(U, V, lam, a, b, D1, D2, dt)
52 u = synth(U);
53 v = synth(V);
54 uuv = u .* u .* v;
55 Un = (U + dt * analys(a - u + uuv)) ./ (1 + (dt * D1) * lam);
56 Vn = (V + dt * analys(b - uuv)) ./ (1 + (dt * D2) * lam);
57end
58```
60Getting from there to the GPU uses numbl for everything up to the IR, and this
61repo only for the backend:
631. **numbl parses and lowers.** Each function is specialized for the concrete
64 argument types of the current grid, via the same `specializeUserFunction`
65 entry point numbl's own JIT uses. Types and array shapes are fixed at this
66 point, so the backend never has to re-decide what an operation means.
672. **numbl's inline pass fuses.** Lowering emits one statement per *operator*
68 (ANF); `inlinePass` folds single-use temps back into their consumer, so one
69 line of MATLAB becomes one expression tree. `uuv = u .* u .* v` arrives as a
70 single statement, not three.
713. **This repo emits WGSL** ([`src/mgpu/wgsl.ts`](src/mgpu/wgsl.ts)). Each
72 element-wise statement becomes one compute kernel that computes one output
73 element per invocation — the WebGPU counterpart of numbl's own C-side fused
74 emitter. Anything it cannot express is refused at compile time with a source
75 position, never silently mis-compiled.
764. **`synth` / `analys` are external operations.** numbl learns their type rules
77 from a `.mtoc2.js` workspace file — its sanctioned extension point for a
78 JS-defined builtin — and the backend maps each call onto the existing
79 spherical-harmonic compute pipelines.
81The Schnakenberg step above compiles to 11 GPU operations: 4 transforms, 5
82generated kernels, and 2 buffer copies feeding the new state back.
84Two consequences worth noting:
86- **The step is synchronous.** WebGPU's encode path (`writeBuffer`, dispatch,
87 `submit`) is all synchronous; only readback and pipeline creation are async, and
88 every pipeline is built once at compile time. So a timestep is pure command
89 recording — the whole batch goes out in one submit, and the only `await` in the
90 loop is the single readback per rendered frame. numbl's own execution being
91 synchronous is therefore not an obstacle: nothing about the algorithm needs to
92 block.
93- **Parameters are uniforms, not constants.** Tunable scalars are deliberately
94 lowered without exact values, so moving a slider rewrites a small buffer
95 instead of triggering a recompile. Editing the MATLAB recompiles; changing `dt`
96 does not.
98## Provenance
100This is the browser port of a MATLAB reference implementation
101(`SphericalReactionDiffusion.m`, "websph"), which defines the solver through a
102four-member porting boundary: `coeffs2vals`, `vals2coeffs`, `grid.lat`,
103`grid.lon`. Profiling of the MATLAB version shows the transforms are ~96% of
104compute, so this port swaps in:
106- **Transforms:** [shtns-webgpu](https://github.com/concept-collection/shtns-webgpu) —
107 fp32 spherical harmonic transforms in WGSL compute shaders, modeled on
108 [SHTNS](https://nschaeff.bitbucket.io/shtns/). Its source is vendored under
109 [`src/sht/`](src/sht/) (CECILL-2.1), including the f64 CPU reference
61e12f1Write the solver in MATLAB and compile it to WebGPUJeremy Magland 110 transform used for testing.
2dedc35turing-sphere: reaction-diffusion on the sphere, spectral solver on WebGPUJeremy Magland 111- **Rendering:** three.js spheres with per-vertex colormaps, adapted from the
112 `SphereEmbedding` view in
113 [figpack](https://github.com/flatironinstitute/figpack)'s experimental
114 extension package ([`src/render/`](src/render/)).
61e12f1Write the solver in MATLAB and compile it to WebGPUJeremy Magland 115- **Solver:** the MATLAB stayed MATLAB. [`models/`](models/) holds the IMEX loop
35d91faDelete the TypeScript solver; the .m models are the only implementationJeremy Magland 116 as `.m` files, executed on the GPU by [`src/mgpu/`](src/mgpu/). There is no
117 second implementation: the app, the desktop benchmark and the tests all compile
118 and run the same `.m`.
35d91faDelete the TypeScript solver; the .m models are the only implementationJeremy Magland 120An earlier version of this repo carried a TypeScript port of the loop alongside
121the `.m`, and used it as the test oracle. That is gone. Two implementations
122agreeing only shows they share assumptions, so the `.m` path is now checked
123against closed-form answers instead — see [Tests](#tests). The one place a second
124implementation is still the right oracle is the transforms themselves, where
125[`src/sht/reference.ts`](src/sht/reference.ts) is shtns-webgpu's own f64
126direct-summation twin.
35d91faDelete the TypeScript solver; the .m models are the only implementationJeremy Magland 128Because the algorithm is compiled to compute shaders, **WebGPU is required** —
129there is no CPU fallback (the f64 CPU transform remains, for tests).
131## Numerics
133- Grid: Gauss–Legendre × equispaced-phi, dealiased for the cubic reactions with
134 the `(pdeg+1)` rule from the reference implementation:
135 `nlat ≥ ((pdeg+1)·lmax+1)/2`, `nphi ≥ (pdeg+1)·lmax+1` (rounded up to a power
136 of two for the GPU FFT path). At the default lmax 63 that is a 128×256 grid.
137- Spectral layout: SHTNS conventions — orthonormal + Condon–Shortley, complex
138 coefficients for m ≥ 0, m-major ordering.
139- fp32 transforms introduce ~1e-6 relative error per step (verified against the
140 f64 CPU path); for pattern formation from 1e-2 seeded noise this is
141 inconsequential.
15a77e2Add a desktop WebGPU benchmark and show its command in the appJeremy Magland 143## Desktop vs browser
145How much does running this in a browser cost? [`scripts/bench.ts`](scripts/bench.ts)
35d91faDelete the TypeScript solver; the .m models are the only implementationJeremy Magland 146runs the *same* thing — same `.m`, lowered by numbl into the same WGSL kernels,
147over the same transforms — from Node on desktop WebGPU (Google Dawn), and the app
148prints the command line that reproduces whatever it is currently simulating:
150```
35d91faDelete the TypeScript solver; the .m models are the only implementationJeremy Magland 151npm run bench -- --preset schnak-spots --lmax 63 --steps 2000 \
15a77e2Add a desktop WebGPU benchmark and show its command in the appJeremy Magland 152 --seed 1 --a 0.1 --b 0.9 --D1 0.0004 --D2 0.008 --dt 0.05
153```
155Copy it from under the stats line, run it, and compare the `ms/step` it reports
156with the app's. Both sides go through the one shared
157[`src/bench/runSpec.ts`](src/bench/runSpec.ts) — the app formats a run into that
158command, the benchmark parses it back — so there is no second copy of the
35d91faDelete the TypeScript solver; the .m models are the only implementationJeremy Magland 159defaults for the two runs to drift apart on. Both then go through the same
160[`ModelSession`](src/mgpu/session.ts), down to the device request in
161`requestShtDevice()` (Dawn is installed under `navigator.gpu` and the WebGPU
162globals, and the rest runs unchanged).
164The benchmark runs under `vite-node`, which is what resolves numbl's compiler
165sources and the `?raw` model imports — plain Node cannot (see
166[The numbl dependency](#the-numbl-dependency)).
168It reports two numbers, because they answer different questions:
170```
171 0.54 ms/step 1857.5 steps/s 92.87 model time/s (batches of 16)
172 one step per submit: 0.74 ms mean · median 0.60 · p05 0.51 · p95 1.29 · min 0.50
173```
175The first is throughput: a batch of steps submitted together and awaited once,
176which is how the app runs and what keeping the state in GPU buffers is for. The
177second is per-step latency, one submit each — comparable to a design that
178synchronises every step, and the only way to get a distribution.
180**What the GPU-resident design is worth.** At lmax 31 on an Intel Xe (Mesa, via
181Dawn) this path runs at **0.25 ms/step**, against **3.01 ms/step** for the
182TypeScript solver this repo used to carry — same machine, same transforms, same
183parameters. A **~12x** difference, and almost all of it is the four per-step
184buffer readbacks that version paid and this one does not. Note that CI, which
185only has a software rasterizer, shows no such gap: there the transforms dominate
186and both designs land within ~10% of each other. The saving is real but it is a
187saving on driver round-trips, so it only appears once the GPU is fast.
e5b7827Fix CI: do not omit optional dependenciesJeremy Magland 189Desktop WebGPU comes from the `webgpu` package (prebuilt Dawn, ~70 MB), listed
190as an optional dependency so that a platform it has no binaries for fails the
191install of that package alone rather than the whole tree. `npm install` picks it
35d91faDelete the TypeScript solver; the .m models are the only implementationJeremy Magland 192up; without it there is no desktop GPU to run on and the benchmark says so.
193Those binaries need glibc 2.29+, which rules out older cluster images
194(RHEL/Rocky 8 is 2.28) unless you run inside a container with a newer base. Other
195flags: `--steps`, `--warmup`, `--batch`, `--json`, `--help`;
196`DAWN_FLAGS='backend=vulkan'` (`;`-separated) passes Dawn options through, e.g. to
197pick a backend or to compare against Dawn's own software adapter.
199What the comparison does and does not control for:
61e12f1Write the solver in MATLAB and compile it to WebGPUJeremy Magland 201- the benchmark is **solver only**; the app's `ms/step` includes the per-frame
35d91faDelete the TypeScript solver; the .m models are the only implementationJeremy Magland 202 readback amortized over its step batch. For a browser number with no rendering,
203 open `test.html?soak=2000&lmax=63`.
15a77e2Add a desktop WebGPU benchmark and show its command in the appJeremy Magland 204- the browser adds its own GPU-process boundary and, for a page that is not
205 cross-origin isolated, coarser timers.
35d91faDelete the TypeScript solver; the .m models are the only implementationJeremy Magland 206- both sides are fp32 throughout, on the same generated kernels, so nothing here
207 is a numerics comparison — only a cost one.
2dedc35turing-sphere: reaction-diffusion on the sphere, spectral solver on WebGPUJeremy Magland 209## Tests
35d91faDelete the TypeScript solver; the .m models are the only implementationJeremy Magland 211There is no second implementation of the solver to diff against, so the `.m` path
212is checked against **closed-form answers**. Each case is one whose evolution is
213known exactly, run through the whole real pipeline — MATLAB source, numbl
214lowering, generated WGSL, GPU transforms — and compared with arithmetic
215([`test/analyticChecks.ts`](test/analyticChecks.ts)):
217- **A** — a linear reaction `f(u) = c*u` leaves every spherical-harmonic mode
218 independent, growing by exactly `(1 + dt*c) / (1 + dt*D*l(l+1))` per step. This
219 pins the transform round-trip, the eigenvalue mapping, the IMEX update and the
220 state feedback at once, and checks that nothing leaks between modes. Agrees to
221 ~2e-7 over 20 steps.
222- **B** — a nonlinear reaction on a *uniform* field stays uniform and diffusion
223 cannot touch it, so each step is exactly the scalar ODE map. Agrees to 1.5e-8
224 over 25 steps. Checks that a generated kernel evaluates a nonlinear reaction.
225- **C** — a 1e-6 perturbation of the Schnakenberg fixed point follows the
226 linearized 2x2 IMEX recurrence, and the `(l=24, m=7)` mode is confirmed
227 unstable. Looser (~2e-3) because fp32 keeps only about four digits of a
228 perturbation that small.
230Two test models exist only for this: [`test/models/linear.m`](test/models/linear.m)
231and [`test/models/logistic.m`](test/models/logistic.m).
233Alongside those, [`test/modelChecks.ts`](test/modelChecks.ts) compiles every model
234the app offers and asserts **how many kernels it compiles to**. That is a fusion
235guard: numbl's lowering emits one statement per *operator* and its inline pass
236folds them back into per-line expression trees, and if that stops happening the
237results stay correct while every operator becomes its own dispatch. It is
238invisible in the numbers, so it is asserted directly. (It has already caught one
239regression.)
241[`test/transformChecks.ts`](test/transformChecks.ts) is the one remaining
242implementation-vs-implementation check, comparing the WGSL transforms against
243shtns-webgpu's f64 CPU twin.
245All three modules run in **both** environments, so the two GPU stacks get the same
246guarantees:
248- `npm run test:node` — under Dawn on the desktop, via `vite-node`. Needs a GPU;
249 pass `--skip-without-gpu` to let a machine without one say so and move on
250 (which is what CI does, since the browser suite covers the same modules).
251- `npm run test:gpu` — builds and drives headless Chrome, on SwiftShader in CI.
252 Also runs the soak.
254Other commands:
256- `npm run bench -- --help` — the desktop benchmark (see
15a77e2Add a desktop WebGPU benchmark and show its command in the appJeremy Magland 257 [Desktop vs browser](#desktop-vs-browser)).
35d91faDelete the TypeScript solver; the .m models are the only implementationJeremy Magland 258- `npx vite-node scripts/longrun-node.ts [lmax]` — run to t = 100 and confirm the
259 pattern saturates into O(1)-contrast spots rather than decaying or diverging.
260- `node scripts/soak.mjs [steps] [lmax]` — drive the demo for many steps,
261 sampling JS heap and catching crashes.
262- `node scripts/screenshot.mjs out.png [light|dark] [minSteps]` — screenshot the
263 demo after a number of steps.
e7bcd70Add soak, live-check and solver-only soak toolingJeremy Magland 264- `node scripts/check-live.mjs [url]` — smoke-check a deployed URL in a real
265 browser: load, press Run, confirm the solver advances.
266- `test.html?soak=<steps>&lmax=<n>` — solver-only soak with no rendering.
268### A note on canvas resizing
270Early long runs killed the browser after ~700–800 steps. The cause was the
271colorbar's min/max labels changing width as their digit count changed, which
272reflowed the panel, fired the `ResizeObserver`, and called
273`renderer.setSize()` — reallocating the WebGL drawing buffer. Assigning
274`canvas.width` also blanks the canvas even when the value is unchanged, so the
275same bug caused visible flicker. Fixed by giving the colorbar column a fixed
276width and making `SphereScene.resize()` return early on no-op resizes.
278## Development
280```
281npm install
282npm run dev # local dev server
283npm run build # type-check + production build to dist/
284```
61e12f1Write the solver in MATLAB and compile it to WebGPUJeremy Magland 286### The numbl dependency
288numbl is a local `file:../../numbl` dependency, so a sibling checkout of
289[numbl](https://github.com/flatironinstitute/numbl) is required. We use its
290compiler internals — parser, lowerer, IR, inline pass — which its package
291`exports` map does not publish, so they are reached through the `numbl-src` path
292alias in [`vite.config.ts`](vite.config.ts).
294The exact surface we depend on is written down in
295[`src/mgpu/numbl.d.ts`](src/mgpu/numbl.d.ts) and TypeScript checks against
296*that*, not against numbl's sources. This keeps this project's compiler settings
297independent of numbl's (its sources do not type-check under the stricter options
298used here), and means a change to one of those shapes upstream breaks the build
299here with a clear diff rather than deep inside numbl's tree.
301The compiler is ~395 kB gzipped and lands in its own chunk. That is the cost of
302compiling MATLAB in the page; a build-time lowering step could remove it at the
303price of no longer being editable live.
305CI clones numbl to the sibling path that `file:` dependency expects, pinned to a
306commit. Two details make that work, both verified by building against a checkout
307that had none of numbl's own dependencies installed:
309- **numbl's `node_modules` are not needed.** The slice we import — parser,
310 lowering, IR, inline pass — is self-contained TypeScript. (Other parts of numbl
311 do import `three`, `react` and `fflate`; we never reach them.)
312- **the install must pass `--ignore-scripts`.** npm runs a linked package's
313 `prepare` script, and numbl's is `husky`, which is not installed in CI.
35d91faDelete the TypeScript solver; the .m models are the only implementationJeremy Magland 315The `scripts/*.ts` entry points that touch the compiler (the benchmark, the node
316tests, the long run) go through `vite-node`, so they resolve imports exactly as the
317browser build does — the `numbl-src` alias and the `?raw` model imports included.
318Plain `node` cannot: numbl's sources import each other as `./foo.js` while the
319files are `.ts`, which needs a bundler's resolution. Scripts that do not touch the
320compiler (`soak.mjs`, `screenshot.mjs`, `check-live.mjs`, `test-gpu.mjs`) are plain
321`.mjs` and run under `node` directly.
2dedc35turing-sphere: reaction-diffusion on the sphere, spectral solver on WebGPUJeremy Magland 323Deployed to GitHub Pages by `.github/workflows/deploy.yml` on push to `main`.
325## License
327CECILL-2.1 (inherited from SHTNS via shtns-webgpu, whose sources are vendored).