Reduction, AD, and numerical scope

SVD parameter directions

svd=true constructs a fixed direction basis before building the augmented layer. This is a nominal reduction, not optimization of the basis alongside the design. The reference computation uses the original, unaugmented problem at problem.p and the supplied algorithm. It differentiates a nominal solve with ForwardDiff, selects params, and includes both state and direct observation sensitivities.

At nominal saved times $t_i$, it stacks all continuous and discrete observation rows without sampling weights and forms the right-endpoint time-sum surrogate

\[F_{\mathrm{ref}}=\sum_{i=1}^{N-1}(t_{i+1}-t_i) J_{\mathrm{ref}}(t_{i+1})^\top J_{\mathrm{ref}}(t_{i+1}).\]

In particular, discrete observations enter this reference as a time-integrated surrogate, not only at their event grids. It is neither the final mixed-design information matrix nor a guarantee of its identifiable directions. It does not use current Lux design parameters, measurement weights, or st.F_init.

For $F_{\mathrm{ref}}=U\operatorname{diag}(\sigma)V^\top$:

1. If ns is omitted, choose the first $n_s$ for which $\sum_{i=1}^{n_s}\sigma_i^2/\sum_i\sigma_i^2\geq\tau_s$, where threshold_singular_values=0.95 by default. A zero reference matrix is rejected in this automatic-selection case.
2. An explicit ns must be an integer between one and length(params).
3. Start with the first $n_s$ left singular vectors. Retain a physical-parameter row only if at least one entry has absolute value greater than threshold_singular_vectors=0.1; set all other rows to zero.

Both thresholds must lie in $[0,1)$. The resulting design.basis is an $n_\theta\times n_s$ matrix. The row truncation is not followed by orthonormalization and does not guarantee full column rank. The selection and basis are fixed construction metadata; no smooth derivative through changing singular-vector selection is claimed.

Sensitivities and single-experiment information use these reduced directions: $G\in\mathbb R^{n_x\times n_s}$ and $F\in\mathbb R^{n_s\times n_s}$. A single-experiment prior must use the same reduced coordinates. For a fixed design with the stated initial-sensitivity assumptions, directional information corresponds to $B^\top F_{\mathrm{physical}}B$ (up to numerical error), using the actual thresholded basis. This is not the full physical information matrix.

See the scientific references for related reduction methods; the steps above specify this implementation rather than all variants in that literature.

Constant mass matrices and residual DAEs

The current interface expects vector states and vector physical parameters. The RHS and observation functions must be compatible with Symbolics tracing. It supports in-place and out-of-place functions; arbitrary numerical callbacks or state-dependent branches are not automatically made symbolically traceable.

For a constant mass matrix,

\[M\dot x=f(x,\theta,v,t),\qquad M\dot G=f_xG+f_\theta B.\]

The augmentation uses one copy of $M$ for the physical state and each sensitivity column, plus an identity block for continuous-information entries. UniformScaling and constant matrix representations are supported. Derivatives of a mass matrix with respect to parameters, state, or time are not added. A singular constant mass matrix still requires a solver and initialization suitable for that DAE.

For a residual-form DAEProblem, write $r(\dot x,x,\theta,v,t)=0$. The implemented sensitivity residual is

\[r_{\dot x}\dot G+r_xG+r_\theta B=0.\]

Continuous information uses the residual $\dot F_c-\sum_m w_mJ_m^\top J_m=0$. When supplied, differential_vars repeats the original flags for each sensitivity column and appends true for each integrated-information entry. When omitted, it remains nothing.

The builder starts from the supplied physical du0, zero augmented derivatives, and a pseudoinverse correction using the initial residual Jacobian with respect to derivatives. This is not a general DAE consistency or initialization algorithm: it cannot fix arbitrary inconsistent algebraic states. Supply consistent physical initial data and a suitable DAE solver. No ModelingToolkit initialization system or arbitrary reset sensitivity map is generated.

Consistency must also hold for the augmented residual. A parameter-dependent algebraic initial state may require nonzero initial sensitivities, which this zero-initial-sensitivity construction does not derive. Consistency of the physical initial state alone does not establish consistency of that augmentation.

Differentiating the design solve

Symbolic augmentation builds numerical equations for $G$ and $F$. Separately, an outer AD backend differentiates the design objective through the numerical augmented solve and parameter reconstruction. Successful symbolic construction or one forward evaluation does not establish backend compatibility.

The OED Core tests exercise ForwardDiff gradients through mixed measurement weights, direct observation dependence, shared-grid multi-experiment parameter containers, and current prior state. They compare selected gradients with finite differences or analytic formulas. The tutorials use AutoForwardDiff() with the shared SciMLBase bridge and Tsit5. ForwardDiff is also used during nominal SVD construction. DAE forward regressions do not establish outer DAE-gradient support.

For an initialized design, ps, and st with nonsingular information, an explicit forward-mode calculation uses the current vector and supplied state:

using ComponentArrays, ForwardDiff
q0, reconstruct = pack(default_vectorizer(), ps)
objective(q, state) = ACriterion()(first(
    fisher_information(design, nothing, reconstruct(q), state)))
gradient = ForwardDiff.gradient(q -> objective(q, st), q0)

This fragment requires the initialized design from a tutorial. It does not differentiate grid times or the SVD selection. A different supplied state, including its current prior, is evaluated by objective(q, state) rather than captured as numerical information during preparation.

The root package's maintained ReverseDiff and Zygote configurations are described in its AD guide. Those root tests do not establish all OED/criterion/DAE combinations. Backend compatibility depends on the solver, sensitivity algorithm, array types, criterion, user callbacks, and mutation. No blanket reverse-mode or higher-order OED support is claimed here.

Schedules, sample indices, grid construction, and saved-layout validation are discrete metadata. Keep them fixed during differentiation and reprepare the optimization model after schedule/layout changes. Numerical parameters, fixed values supplied in state, and current priors remain in the computation; avoid copied numerical snapshots or stale reconstruction closures. The ownership guide explains why mutating a trajectory's retained containers can change later information queries.