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.
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.
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.
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
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
110 transform used for testing.
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/)).
115- **Solver:** the MATLAB stayed MATLAB. [`models/`](models/) holds the IMEX loop
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`.
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.
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.
143## Desktop vs browser
145How much does running this in a browser cost? [`scripts/bench.ts`](scripts/bench.ts)
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```
151npm run bench -- --preset schnak-spots --lmax 63 --steps 2000 \
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
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.
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
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.
199### Comparing the two honestly
201The app reports **two** numbers, and only the first is comparable to the
202benchmark:
204```
205solver 0.58 ms/step (1724 steps/s) · 12.4 ms/frame incl. readback + render
206```
208`solver` is the batch of steps alone, waited for but not read back — the same
209thing the benchmark's throughput number measures. `ms/frame` additionally carries
210a GPU→CPU readback **per species**, the colormapping, and the vertex upload.
212Those per-frame costs are fixed: they do not shrink when the GPU gets faster. So
213the faster your GPU, the larger the ratio between them — on a quick discrete GPU
214it is easy for a frame to cost ten times the four steps inside it, purely because
215a `mapAsync` round trip in a browser has to drain the queue and cross into the GPU
216process. **That is expected, and it is not the solver being slower in the
217browser.** Compare `solver` with the benchmark's throughput line; comparing
218`ms/frame` against it measures the readback, not the computation.
220Other things the comparison does not control for:
222- the browser's renderer→GPU-process boundary on every submit, where Dawn in Node
223 is in-process; and, for a page that is not cross-origin isolated, coarser
224 `performance.now()`.
225- both sides are fp32 throughout, on the same generated kernels, so nothing here
226 is a numerics comparison — only a cost one.
228### Why the browser is slower, and how to find out by how much
230Some gap is real and some is measurement. Four numbers, in increasing order of
231what they include — walk down them and the gap attributes itself:
233| number | includes |
234|---|---|
235| `npm run bench -- --lmax 63` | desktop solver: batched steps, one sync per batch, in-process Dawn |
236| `test.html?soak=2000&lmax=63` → `solver` | browser solver: same batching, no rendering at all |
237| the app's `solver` | browser solver, measured in a periodic batch of 32 |
238| the app's `ms/frame` | four steps **plus** a readback per species, colormapping and the vertex upload |
240If the soak matches the benchmark, the solver is fine in the browser and
241everything above it is readback and rendering. If the soak is itself slower, the
242remaining suspects are:
244- **the GPU-process boundary.** Every submit and every sync is IPC out of the
245 renderer; Dawn in Node is in-process. This is a fixed per-batch cost, so it hurts
246 most when the GPU is fast. `npm run bench -- --batch 4` makes the desktop pay a
247 sync as often as the app's frame loop does, which shows how much of the gap is
248 just amortization.
249- **competing with the renderer.** The page draws two spheres through WebGL on the
250 same GPU, in its own animation loop. The soak has no renderer, so comparing the
251 soak against the app's `solver` separates contention from everything else.
252- **clocks.** An animation-paced loop leaves the GPU idle for most of each 16 ms
253 frame, so it may never leave its low-power state, while the benchmark hammers it
254 continuously and boosts. On a thermally managed laptop this alone can be worth a
255 factor of two, and it is not something the code can fix.
256- **which browser.** WebGPU implementations differ substantially in maturity;
257 Chrome and Safari are not interchangeable for this.
259None of these change *what* is computed — see below for how to confirm that
260independently.
262### Is it really the same computation?
264```
265node scripts/compare-env.mjs [--lmax 31] [--steps 200] [--preset schnak-spots]
266```
268runs one identical spec on the desktop and in a real browser and compares the
269final spectral state. The pipeline is deterministic given (model source,
270parameters, lmax, seed, steps) — a seeded PRNG, then fixed arithmetic — so the two
271should agree to fp32 round-off. Both sides build their spec through the same
272`parseArgs`, so neither can quietly use a different default.
274They will *not* agree bit for bit; GPUs differ in fused-multiply-add and other
275latitude fp32 allows. Between Intel Xe (via Dawn) and SwiftShader — about as
276different as two implementations get — 200 steps at lmax 31 agree to a relative
277L2 of **2e-6**.
279It also reports which **Fourier stage** each side chose. `ShtPlan` picks FFT or
280DFT from the device's workgroup-storage and invocation limits, and those are
281genuinely different algorithms that round differently, so a mismatch there
282explains a difference in the values rather than being a symptom of one. The app's
283stats line and the benchmark both print the chosen stage for the same reason.
285## Tests
287There is no second implementation of the solver to diff against, so the `.m` path
288is checked against **closed-form answers**. Each case is one whose evolution is
289known exactly, run through the whole real pipeline — MATLAB source, numbl
290lowering, generated WGSL, GPU transforms — and compared with arithmetic
291([`test/analyticChecks.ts`](test/analyticChecks.ts)):
293- **A** — a linear reaction `f(u) = c*u` leaves every spherical-harmonic mode
294 independent, growing by exactly `(1 + dt*c) / (1 + dt*D*l(l+1))` per step. This
295 pins the transform round-trip, the eigenvalue mapping, the IMEX update and the
296 state feedback at once, and checks that nothing leaks between modes. Agrees to
297 ~2e-7 over 20 steps.
298- **B** — a nonlinear reaction on a *uniform* field stays uniform and diffusion
299 cannot touch it, so each step is exactly the scalar ODE map. Agrees to 1.5e-8
300 over 25 steps. Checks that a generated kernel evaluates a nonlinear reaction.
301- **C** — a 1e-6 perturbation of the Schnakenberg fixed point follows the
302 linearized 2x2 IMEX recurrence, and the `(l=24, m=7)` mode is confirmed
303 unstable. Looser (~2e-3) because fp32 keeps only about four digits of a
304 perturbation that small.
306Two test models exist only for this: [`test/models/linear.m`](test/models/linear.m)
307and [`test/models/logistic.m`](test/models/logistic.m).
309Alongside those, [`test/modelChecks.ts`](test/modelChecks.ts) compiles every model
310the app offers and asserts **how many kernels it compiles to**. That is a fusion
311guard: numbl's lowering emits one statement per *operator* and its inline pass
312folds them back into per-line expression trees, and if that stops happening the
313results stay correct while every operator becomes its own dispatch. It is
314invisible in the numbers, so it is asserted directly. (It has already caught one
315regression.)
317[`test/transformChecks.ts`](test/transformChecks.ts) is the one remaining
318implementation-vs-implementation check, comparing the WGSL transforms against
319shtns-webgpu's f64 CPU twin.
321All three modules run in **both** environments, so the two GPU stacks get the same
322guarantees:
324- `npm run test:node` — under Dawn on the desktop, via `vite-node`. Needs a GPU;
325 pass `--skip-without-gpu` to let a machine without one say so and move on
326 (which is what CI does, since the browser suite covers the same modules).
327- `npm run test:gpu` — builds and drives headless Chrome, on SwiftShader in CI.
328 Also runs the soak.
330Other commands:
332- `npm run bench -- --help` — the desktop benchmark (see
333 [Desktop vs browser](#desktop-vs-browser)).
334- `npx vite-node scripts/longrun-node.ts [lmax]` — run to t = 100 and confirm the
335 pattern saturates into O(1)-contrast spots rather than decaying or diverging.
336- `node scripts/soak.mjs [steps] [lmax]` — drive the demo for many steps,
337 sampling JS heap and catching crashes.
338- `node scripts/screenshot.mjs out.png [light|dark] [minSteps]` — screenshot the
339 demo after a number of steps.
340- `node scripts/check-live.mjs [url]` — smoke-check a deployed URL in a real
341 browser: load, press Run, confirm the solver advances.
342- `node scripts/compare-env.mjs` — run one identical spec on the desktop and in a
343 browser and compare the final state (see
344 [Is it really the same computation?](#is-it-really-the-same-computation)).
345- `test.html?soak=<steps>&lmax=<n>` — solver-only soak with no rendering.
347### A note on canvas resizing
349Early long runs killed the browser after ~700–800 steps. The cause was the
350colorbar's min/max labels changing width as their digit count changed, which
351reflowed the panel, fired the `ResizeObserver`, and called
352`renderer.setSize()` — reallocating the WebGL drawing buffer. Assigning
353`canvas.width` also blanks the canvas even when the value is unchanged, so the
354same bug caused visible flicker. Fixed by giving the colorbar column a fixed
355width and making `SphereScene.resize()` return early on no-op resizes.
357## Development
359```
360npm install
361npm run dev # local dev server
362npm run build # type-check + production build to dist/
363```
365### The numbl dependency
367numbl is a local `file:../../numbl` dependency, so a sibling checkout of
368[numbl](https://github.com/flatironinstitute/numbl) is required. We use its
369compiler internals — parser, lowerer, IR, inline pass — which its package
370`exports` map does not publish, so they are reached through the `numbl-src` path
371alias in [`vite.config.ts`](vite.config.ts).
373The exact surface we depend on is written down in
374[`src/mgpu/numbl.d.ts`](src/mgpu/numbl.d.ts) and TypeScript checks against
375*that*, not against numbl's sources. This keeps this project's compiler settings
376independent of numbl's (its sources do not type-check under the stricter options
377used here), and means a change to one of those shapes upstream breaks the build
378here with a clear diff rather than deep inside numbl's tree.
380The compiler is ~395 kB gzipped and lands in its own chunk. That is the cost of
381compiling MATLAB in the page; a build-time lowering step could remove it at the
382price of no longer being editable live.
384CI clones numbl to the sibling path that `file:` dependency expects, pinned to a
385commit. Two details make that work, both verified by building against a checkout
386that had none of numbl's own dependencies installed:
388- **numbl's `node_modules` are not needed.** The slice we import — parser,
389 lowering, IR, inline pass — is self-contained TypeScript. (Other parts of numbl
390 do import `three`, `react` and `fflate`; we never reach them.)
391- **the install must pass `--ignore-scripts`.** npm runs a linked package's
392 `prepare` script, and numbl's is `husky`, which is not installed in CI.
394The `scripts/*.ts` entry points that touch the compiler (the benchmark, the node
395tests, the long run) go through `vite-node`, so they resolve imports exactly as the
396browser build does — the `numbl-src` alias and the `?raw` model imports included.
397Plain `node` cannot: numbl's sources import each other as `./foo.js` while the
398files are `.ts`, which needs a bundler's resolution. Scripts that do not touch the
399compiler (`soak.mjs`, `screenshot.mjs`, `check-live.mjs`, `test-gpu.mjs`) are plain
400`.mjs` and run under `node` directly.
402Deployed to GitHub Pages by `.github/workflows/deploy.yml` on push to `main`.
404## License
406CECILL-2.1 (inherited from SHTNS via shtns-webgpu, whose sources are vendored).