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Analytical sparse Jacobians for EarthSciAST
(.esm) models, by symbolic differentiation of the expression AST.
Given an arbitrary .esm model (or a coupled document's flattened system),
EarthSciASTDiff computes ∂f/∂u (states), ∂f/∂p (parameters, array-valued
ones cell by cell), ∂f/∂B (live forcing buffers) and ∂f/∂t of the ODE
right-hand side as a list of structured bands — rectangular index
regions with symbolic column-index expressions and ESM coefficient
expressions. From the bands it derives, without any numeric probing:
the exact structural sparsity pattern (global: ifelse/min/max
branches unioned, so it is valid at every point),
a structure classification (:block_diagonal / :banded /
:general, and :column_block along a column axis the caller names) for
choosing a matrix type,
optional symbolic factorization info (general interface; per-cell
block LU with Markowitz ordering and fill-in — the chemistry shape — is
the first implementation),
a serialized "jacobians" document block, so the Jacobian is itself
an ESM artifact any language binding can evaluate,
product documents: J's band fields, J v, Jᵀ λ and the
second-order contraction, each as a derived ESM document an unmodified
EarthSciAST evaluator runs,
second-order bands for Rosenbrock discrete adjoints, contracted to
∇(μᵀ J w) per step without ever forming a Hessian.
Every binding manipulates ASTs only: every number comes from EarthSciAST
evaluating one of its documents, and
EarthSciIntegrator
coordinates those evaluations in a solver. All three bindings emit the
evaluation and product documents and the scatter plans, and every binding's
value tests get their numbers by running those documents through
EarthSciAST.
Correctness contract: the derivative is taken on the same expanded,
lowered tree EarthSciAST.build_evaluator compiles, and the test suite pins
the assembled Jacobian against ForwardDiff through the tree-walk RHS to
machine precision.
Quick start (Julia)
using EarthSciAST, EarthSciASTDiff
file = EarthSciAST.load_path("model.esm")
entries = jacobian_bands(file, "MyModel") # the Jacobian as structured bands
pat = jacobian_pattern(file) # structural pattern + row/column names
detect_structure(entries, sysview(file, "MyModel")) # :block_diagonal | :banded | :general
doc = jacobian_document(file, "MyModel") # document + "jacobians" block
# Values are EarthSciAST's to compute. Each product is a document:
ev, fields = evaluation_document(file) # J's band fields (model "JacobianEval")
jv, io = jvp_document(file) # J v (model "JvpEval")
vj, io = vjp_document(file) # Jᵀ λ (model "VjpEval")
hv, io = hvp_document(file; stages = 2) # ∇_u Σᵢ μᵢᵀ J(u) wᵢ (model "HvpEval")
# Build any of them with EarthSciAST, write u into the source states' slots
# and the inputs (io.v, io.lam, io.mu / io.w) into theirs, call the RHS once,
# and read the outputs (io.jvp, io.vjp, io.hvp) from du.
f!, x0, p, _, vm = build_evaluator(EarthSciAST.load_document(jv); model_name = io.model)
# Coupled documents go through flatten first. The evaluation and product
# documents of a flattened system declare its scoped names (`Chem.O3`) in
# the component tree (models `Chem`, …), so EarthSciAST loads them; the band
# fields are `JacobianEval.J_k`, the inputs and outputs unscoped in their
# model (`JvpEval._v_Chem_O3`; spec §8.1, §9.1).
flat = flatten(file)
ev, fields = evaluation_document(flat)
jv, io = jvp_document(flat)
∂f/∂p: pass wrt = :parameters. ∂f/∂B, with respect to the cells of
the forcing buffers (by default every array parameter with a data update;
pass forcing = [names…] to choose), is wrt = :forcing:
vjp_document(file; wrt = :forcing) is the forcing cotangent
ḡ_B = (∂f/∂B)ᵀ λ, its buffers bound live through EarthSciAST's
param_arrays. ∂f/∂t is wrt = :time.
Second order, for a Rosenbrock adjoint: an axis pair (a1, a2),
a1 ∈ (:states, :time), a2 ∈ (:states, :parameters, :forcing):
In jacobian_document a single pair block is wrt = :states => :states; a
bare tuple there is refused, since it also reads as a list of axes.
A Jᵀ λ or second-order document is an equality-filtered faq scatter
(spec §9.3): EarthSciAST.jl's interpreter evaluates it today
(compiler = :interpreter), and its native compiler gains it with the
equality-filter scatter optimization (EarthSciAST#613).
Placing J's band values in a matrix of your own layout is a scatter plan:
scatter_plan(file, Csr()) (or LaneBlocks(), LaneSparseBlocks(),
ColumnBanded(axis)), checked against the source by bind_plan, whose
pairs say which JacobianEval field cell goes to which storage position.
Documentation
The documentation site — getting-started guide, cross-language API reference and the
specification — is at https://earthsci.dev/astdiff/. It is built from docs/ by
the shared tooling in EarthSciML/earthsci.dev
(.github/workflows/docs.yml publishes it on every push to main); preview locally
with python3 <earthsci.dev>/tools/api_docs/render.py --repo-root . (see that repo's
tools/api_docs/README.md).
Dependency on EarthSciAST
Every binding depends on a range of published EarthSciAST releases, never
on a path or a git branch:
Binding
Manifest
Range
Julia
pkg/EarthSciASTDiff.jl/Project.toml[compat]
EarthSciAST = "0.6, 0.7, 0.8, 0.9"
Rust
pkg/earthsci-astdiff-rs/Cargo.toml
earthsci-ast = ">=0.5, <0.10"
Python
pkg/earthsci-astdiff-py/pyproject.toml
earthsci-ast>=0.4,<0.10
CI tests what those ranges promise:
Conformance — the release gate.
Each binding runs against the lowest and the latest published
release inside its range, so the floor is tested rather than assumed and
the latest is what a fresh install gets.
scripts/earthsci-ast-versions.py
reads the manifests and the registries (General, crates.io, PyPI) to pick
them, and a binding whose two ends coincide runs once. The Python binding
runs each end on both its requires-python floor and the newest Python. A
failed weekly run opens an issue: it is the run that sees a new
EarthSciAST release inside a range, which fresh installs already get.
Upstream (EarthSciAST main) — an
early warning. The same suites run against EarthSciAST main, checked out
as a sibling, on every PR and weekly; a failed weekly run opens an issue.
It does not gate a release. When main has bumped its version ahead of a
release, the run widens the Julia and Cargo ranges for itself only
(earthsci-ast-versions.py --admit-checkout); never commit such a widening.
Both workflows run the same suites, defined once in
bindings.yml.
A change that needs an EarthSciAST feature not yet released therefore passes
the upstream check and fails conformance. The order is: release EarthSciAST,
raise the floor here, then release this package. Upper bounds move when a new
EarthSciAST minor release passes (Dependabot proposes the bump).
To test locally against an EarthSciAST checkout at ../EarthSciAST: Julia
Pkg.develop(path="../EarthSciAST/pkg/EarthSciAST.jl") in a scratch
environment; Python picks it up automatically when earthsci_ast is not
installed (tests/conftest.py); Rust
cargo test --config 'patch.crates-io.earthsci-ast.path="../../../EarthSciAST/pkg/earthsci-ast-rs"'.
Specification
The "jacobians" block format, the per-operator derivative semantics
(including the non-smooth policies: branch-union sparsity, max/min tie
rules, closed-function table), and the factorization-info interface are
specified in esm-jacobian-spec.md — a draft to be
merged into esm-spec.md once stable. Cross-language conformance fixtures
and goldens live in tests/.
Repository layout
Mirrors EarthSciAST to make room for future language bindings:
Planned bindings follow EarthSciAST's naming: earthsci-astdiff-py,
earthsci-astdiff-rs, earthsci-astdiff-go, earthsci-astdiff-ts — each
implements the §3 rule table and band calculus against its language's
EarthSciAST implementation and is gated by the same goldens.
Design notes
Bands, not a rewritten array AST. A band is the symbolic analogue of
the tree-walk's _AccDesc state-read descriptors: col == row is a
per-cell (chemistry) coupling, col == row ± k an affine stencil offset.
Structure is read off the column expressions.
Sparsity is structural occurrence — never a symbolic-zero or numeric
probe (the Symbolics/KPP/AMICI/OpenModelica consensus). Stored zeros are
kept so solver preparations can be reused across the integration.
Evaluation is EarthSciAST's. Band coefficient fields and every
product (spec §9) are emitted as derived ESM documents evaluated by the
unmodified EarthSciAST evaluator; this package computes no values, so a
faster evaluator is an EarthSciAST change, never a kernel here.
Factorization info is a general interface (FactorizationPlan,
dispatched on the detected structure class); BlockDiagonalPlan
(per-cell block LU: Markowitz ordering + symbolic fill-in, the KPP
precedent) is the first implementation. Banded and general-CSC plans slot
in without changing the interface.
Roadmap
Coefficient-size control, stage 1: branch-aware simplification
(simplify_branches — equal-branch collapse, same-condition pruning)
in the band calculus, plus hash-consed template CSE at emission
(cse_templates → zero-param expression_templates in the block,
spec §4.1).
Coefficient-size control, stage 2 (PPM-scale): CSE inside the
evaluation model (hoist_observed — repeated coefficient subtrees
become factored observed-array buffers of the derived document,
evaluated once per cell per RHS call). PPM: derived document 4×
smaller, jac! 8× faster per call (beating ForwardDiff through the
RHS by an order of magnitude); values bit-identical to the plain
path. Symbolic reverse mode turned out unnecessary at this scale —
the remaining prepare_jacobian cost is tree-walk kernel JIT, i.e.
the dedicated compiled jac! kernel already noted in Design notes.
Trimmed evaluation document: the source's state derivatives are
zeroed in the derived model, so a jac! call evaluates the band
fields and the observed they read but no longer recomputes f
(6–36% less time per jac! call on the fixtures, copy-in and scatter
included; the saving grows with the cost of f). Pinned by
*.evaldoc.json.
Scatter plans (spec §8): a checked, serializable map from the band
fields to a consumer's storage — lane-SoA dense per-cell blocks
(Markowitz block_vars order), KPP-style sparse lane blocks, per-column
bands and CSR. Lane layouts are per-cell templates (O(nnz)); the lane
capacity is a bind-time choice. bind_plan refuses a plan from another
band list or state set, and jac_into! fills the layout from a prepared
evaluator. Pinned by *.<layout>.plan.json, which Rust reproduces.
Static region clipping for index(makearray…) reads (clip.jl):
decidable membership guards are decided exactly by splitting band row
ranges at their affine breakpoints (equal-coefficient sub-bands
re-merge), so region structure never reaches the coefficients or
pads the pattern; state-dependent (limiter) branches remain unioned.
Contracted-index bands: reduce: "+" faq nodes over gathered
columns (u[j], u[i+k], u[conn[i,k]] const tables) emit entries
with free column dimensions (accumulating at assembly);
contraction-independent columns emit ordinary entries with a
symbolic-sum coefficient.
interp.bilinear derivative (two clamped-weight partials, per-cell
slope constants precomputed into nested segment ifelse, spec §3.4)
and tgrad (wrt = :time through the same band calculus;
ode_components/odefunction attach the analytical tgrad!, with a
zero fill for autonomous systems).
Time-derivative bands specified and golden-gated
(#11):
wrt: "time" is spec §4.2 (one scalar column t, every read observed
inlined, the partial with parameters held fixed, data-borne time
dependence excluded, an empty list meaning f_t ≡ 0), and
tgrad_photolysis.time.{bands,block,evaldoc,jvals}.json pin Julia,
Rust and Python against each other. datetime.julian_day now
differentiates to 1/86400 (it is continuous; the other datetime.*
fields stay 0), and the Rust and Python golden writers emit floats in
the reference's layout (2.4405875e6, not 2440587.5).
Forcing-buffer and array-parameter columns
(#1, spec
§4.3): wrt: "forcing" over the cells of the live data buffers (the
forcing set inferred from data updates or passed in), the record
bracket an ordinary leading length-2 axis so the time blend's chain rule
falls out of */+; array parameters are real columns of
wrt: "parameters" instead of being dropped at assembly. Blocks carry
their columns (the canonical column ordinal) and a model's
"jacobians" entry may be an array of blocks. The forcing cotangent of
a stage is the vjp product document on this axis (spec §9).
Pinned by forcing_bracket.{forcing,parameters}.* goldens.
Second-order bands for Rosenbrock discrete adjoints
(#2, spec
§4.4): the band calculus applied to first-order coefficients, emitted
as wrt: [a1, a2] blocks (col2, col2_idx), and contracted per
adjoint step by the hvp product document (g[col2] += coef2 · Σᵢ μᵢ[row] wᵢ[col], or
the symmetric form on the state pair) — checked against ForwardDiff
over ForwardDiff. Pinned by <fixture>.<a1>-<a2>.* goldens on
bd_chem, arrhenius_chem, forcing_bracket and minmod_adv.
Python and Rust reproduce the band and block goldens byte-for-byte
and the contraction values through their hvp product documents.
Filtered faq nodes, max/min/* reductions and value-indexed reads
(#3, spec
§3.5): a filter is a gate whose Dirac term is dropped; its index-only
conjuncts split the band box exactly and leave a triangular window as
the entry's where guard (a level scan's gk < gke), its value
conjuncts stay in the coefficient. max/min weight each term by the
first-extremum indicator (the left-fold tie rule), products by the
exclusion product (no division). A subscript that reads a value (a
level search x[κ(u)]) is one-hot over the axis. detect_structure(…; column_axis) reports :column_block with a line-by-line
factorization plan. Pinned by filtered_scan and semiring_reduce
goldens.
Causal recurrences (#3,
spec §4.5, §9.7) — Rust and Python first, the Julia reference once
EarthSciAST.jl runs recurrences (EarthSciAST#546): an observed that
reads itself at an earlier cell is an intermediate, differentiated
on its definition and as an extra column of its readers. The block
composes the chain rule through the line's transfer recurrence
T = (I − G)⁻¹ (the block's auxiliary); the product documents run
tangent and adjoint sweeps instead (the adjoint on the reversed
axis), and the second-order contraction runs both plus one more
adjoint sweep. column_block plans gain diag_plus_rank1 (a
column-mean mixing term), and a where guard may gather from a
const mask. Pinned by recurrence_sweep and pbl_rank1 goldens the
Rust binding writes (tests/write_goldens.rs), checked against
central differences of the source right-hand side.
Public expansion seam in EarthSciAST (EarthSciAST.expanded_model,
≥ 0.9.1) and ODEFunction(f!; jac, jac_prototype) / ODEProblem
wiring (odefunction / odeproblem via the SciMLBase package
extension; ode_components is the solver-free half).
Python binding, band-calculus half (pkg/earthsci-astdiff-py):
the §3 rule table, observed inlining, static clipping, contracted
bands, and the §4 serialization — byte-identical to the Julia
reference on every fixture × axis (goldens now cover all five
fixtures). Includes a faithful port of the reference simplify
(the two packages' own simplifiers order commutative arguments
differently). Its tests check values against the Julia-produced VALUE
goldens tests/goldens/*.jvals.json — exact pattern key set, 1e-10
relative values — by running its documents through EarthSciAST's
Python interpreter, as Julia's do (tests/docvals.py). The emission side is in too —
jacobian_block with §4.1 template CSE and the block-LU
factorization plan, byte-identical to the reference's blocks
(*.block.json goldens). Emits the evaluation document (with
hoist_observed) and the jvp / vjp / hvp product documents
byte-identically to Julia, each product checked by value by
EarthSciAST's Python interpreter against the value goldens, and the
scatter plans (scatter.py, *.plan.json), each bound plan checked
by scattering the evaluated band fields.
Rust binding (pkg/earthsci-astdiff-rs), same scope as Python:
band calculus, block emission (template CSE + structure + block-LU
plan), and value tests run through its documents
(tests/common/docvals.rs) — all three golden gates pass
byte-for-byte / at tolerance on every fixture. Caught up with esm
1.0.0/1.1.0 (the faq tag, the unknown/parameter variable model,
inline pruning) together with Python. Emits the derived evaluation
document (with hoist_observed) byte-identically to Julia
(*.evaldoc.json) and the scatter plans (*.plan.json), with
each bound plan's scatter of the evaluated band fields checked
against J on every fixture, and the jvp / vjp / hvp product documents
(products.rs), each checked by value by earthsci-ast-rs's
interpreter against the value goldens.
Production-rate Rust assembly
(#4 W2, W4,
W5) — since retired: JacobianProgram / HvpProgram compiled the
evaluation document through earthsci-ast-rs and scattered, applied
and contracted the band slab here. That execution is the consumer's
(EarthSciIntegrator compiles the same evaluation_document and runs a
bound ScatterPlan); examples/bench_production.rs times
EarthSciAST evaluating the document.
diffsol solver integration — not needed by EarthSciIntegrator, which
owns its linear algebra; it could be built on the Csr layout.
