There is a design trick, used without fanfare in analog-computing constructions, that deserves a name. An intermediate quantity may appear in the reasoning that produces an ODE — a piece of algebra that motivates why the equations look the way they do — and then vanish before the ODE is written down. It lives in the derivation but not in the execution. The paper Bounded Analog Complexity by Chen and Huang calls this the pass-through technique (Remark 6.3).

This post pulls the trick out of its remark, writes down what it means, asks when it is applicable, and connects it to two pieces of older and newer machinery: Kalman’s minimal-realization theory from the 1960s, and SINDy-AE from 2019. At the end there is a small numerical experiment — running sparse regression on a construction where the pass-through has already been performed — whose outcome is informative in an unexpected way.

1. Where the trick shows up

The $\alpha^{\beta}$ construction in Bounded Analog Complexity computes real-number exponentiation as an ODE. The design goes through an exponent $R = y \ln(1+x_1)$, so that the output $z$ satisfies $z = e^{R}$ and $\dot{z} = z \dot{R}$. Differentiating:

The right-hand side contains a logarithm and a reciprocal — neither allowed in a polynomial ODE. The construction finesses this by introducing two tracked helper variables with their own clean polynomial dynamics,

so that $\dot{R}$ becomes polynomial in $(x_1, u, v, y, \dot{y}, \dot{x}_1)$. Substituting:

This is the equation the paper prints. Notice what happened to $R$. It appeared in the derivation as scaffolding. It is absent from the final system. The ODE tracks $(x_1, u, v, Z)$ and upstream inputs; nothing else. $R$ existed long enough to suggest the right form, and then was gone.

A similar thing happens elsewhere in the same paper. In the bounded surrogate compilation (Construction 3.1), the physical time $t$ and a possibly-unbounded driver $f$ appear in the derivation but do not appear as state variables of the bounded surrogate system. They are passed through.

The pattern is the same: a helper variable, which may even be non-polynomial in the tracked states, shows up in the argument that establishes correctness and then does not appear in the ODE the machine runs.

2. What pass-through means, precisely

The informal remark can be stated cleanly once we separate two roles a variable can play. Let

be a polynomial ODE on $(Z, v)$, where $Z = (z_1, \ldots, z_n)$ and $v = (v_1, \ldots, v_m)$. Fix a subset $O \subseteq \{z_1, \ldots, z_n\}$ and call it the set of observables — the variables whose trajectories we care about. The helper $v$ is pass-through with respect to $O$ if there exists a polynomial ODE

whose trajectories agree exactly with those of the original system on $O$, for every initial condition. In words: $v$ can be removed without loss of observable behavior, and the removal leaves a polynomial system.

The subtlety is that the original ODE does syntactically depend on $v$; the claim is that once $\dot{v}$ is substituted in terms of $Z$, the resulting equations can be closed within $Z$ alone.

3. A sufficient condition

Two conditions, together, force pass-through:

(i) There is a polynomial map $F: \mathbb{R}^{n} \to \mathbb{R}^{m}$ such that along every trajectory $v(t) = F(Z(t))$.

(ii) There is a polynomial map $G: \mathbb{R}^{n} \to \mathbb{R}^{m}$ such that $\dot{v}(t) = G(Z(t))$.

If both hold, substitute $v = F(Z)$ into $p(Z, v)$ and the reduced system $\dot{Z} = p(Z, F(Z))$ is polynomial in $Z$ and its trajectories agree with the original on $Z$.

The $\alpha^{\beta}$ case is actually outside this strict scope — $R = y \ln(1+x_1)$ is not polynomial in the other states. What rescues it is that $\dot{R}$ is polynomial, and the observable is $z = e^{R}$ rather than $R$ itself. So condition (i) is replaced by the weaker requirement that $\dot{z} = z \dot{R}$ closes polynomially. The general question of when elementary-but-not-polynomial helpers admit pass-through is open; a polynomial-helper decision procedure, on the other hand, reduces to a Gröbner-basis consistency check on the ideal

4. Pass-through under two older names

The notion is not new to analog computing. What is new is the explicit use of pass-through by a designer, with observables chosen and algebra performed by hand.

Realization theory. Given an input–output map $u(t) \mapsto y(t)$, Kalman’s minimal-realization theorem asks for the smallest-dimensional state space that reproduces it. In the nonlinear case, Hermann and Krener (SIAM J. Control, 1977) decompose the state space along an observability foliation and quotient out the unobservable directions. The observable quotient is a pass-through elimination. What the designer of a bounded GPAC does by hand, the Hermann–Krener machinery does in principle on any smooth input-affine system.

SINDy-AE. The paper of Champion, Lusch, Kutz, and Brunton (PNAS 116, 2019) learns an autoencoder $\phi: \mathbb{R}^{n} \to \mathbb{R}^{k}$ together with a sparse polynomial ODE on the latent variables $\widetilde{Z} = \phi(Z)$, jointly minimizing reconstruction loss and dynamics residual. If one fixes $\phi$ to be projection onto the observables $O$ and demands that the latent ODE reproduce trajectories on $O$ exactly, what SINDy-AE learns is — up to numerical tolerance — the pass-through reduction. SINDy-AE is the data-driven counterpart of the hand derivation: the hand method produces an exact algebraic identity, the data-driven method a sparse regression.

The two-pole picture is clean. Hand: symbolic ODE in, algebraic elimination, reduced polynomial ODE out. Data-driven: trajectory samples in, sparse latent regression, reduced polynomial ODE out. Same output, different inputs.

5. A stress test

A natural question is whether ordinary SINDy, without an autoencoder, can recover pass-through-reduced constructions from trajectory data. A positive answer would say: the hand work done by the designer of $\alpha^{\beta}$ is exactly what SINDy does when fed its own output.

The test setup: take the 10-dimensional reduced system for $(\pi/4)^{e}$ from Chen and Huang’s construction, generate trajectories from $100$ perturbed initial conditions, run STLSQ-based SINDy with a degree-$4$ monomial library (size $\binom{14}{4} = 1001$), and see how many of the ten equations are recovered with their true sparsity pattern. Perturbations are $\sigma = 0.1$ on the six upstream inputs and $\sigma = 0.5$ on the four downstream variables, large enough to break the strong transient collinearity the downstream block exhibits at canonical initial conditions.

The result divides cleanly in two.

Seven of the ten equations are recovered exactly: the six upstream equations that prepare the inputs $\alpha$ and $\beta$ (each with one or two true terms, zero false positives, coefficient error below $10^{-4}$), and the single equation $\dot{x}_1 = z_a - 1 - x_1$ with all three of its true terms and nothing else.

Three equations — the downstream ones for $(u, v, Z)$ — fail catastrophically. Each retains every true term (recall is $1$), but every one of them also retains roughly $1000$ false-positive terms with wild coefficients. STLSQ cannot prune.

6. Why the failure is the interesting part

The natural first read is that SINDy is under-tuned. A higher threshold $\tau$, more regularization, more data, better perturbations — some knob should close the gap.

None of them do, and the reason is structural. The downstream block of the construction has the cascade

and the $x_1$ equation pulls $x_1$ toward $z_a - 1$ exponentially fast. Once that has happened, the factor $(z_a - 1 - x_1)$ is small along the whole remaining trajectory, and the monomials $x_1$ and $z_a$ are fused by a near-algebraic identity on the attractor submanifold.

That identity sits in the degree-$4$ monomial library as a near-zero linear combination of basis monomials. The regression matrix columns for $\dot{u}, \dot{v}, \dot{Z}$ are correspondingly rank-deficient. A true-sparsity solution and a true-sparsity-plus-null-space solution have essentially the same residual. Thresholded least squares, handed a near-singular system, retains every candidate column because no single one is large enough to survive the threshold against a null-space coordinate. That is the full story of the three failing equations.

This is the same algebraic relation that makes the hand-derived pass-through possible. The fact that $x_1 \to z_a - 1$ is what lets the designer close $\dot{R}$ polynomially without tracking $R$; it is also what lets SINDy see a lower-dimensional attractor without being able to name it. The hand method performs the elimination explicitly; SINDy detects the latent algebraic structure as regression rank deficiency, and lacks the symbolic machinery to complete it.

So the experimental result is not “SINDy fails on $\alpha^{\beta}$.” It is:

Whenever a construction relies on pass-through for its polynomial form, ordinary SINDy sees the relevant algebraic identity as a rank-deficient regression, not as an eliminable variable. The information is there; the format is wrong.

This is exactly the gap that SINDy-AE is built to cross: introduce a lift $\phi$, parameterize the attractor submanifold explicitly, and sparse regression on the lifted coordinates should close. That is the next experiment to run.

7. Remarks

The pass-through technique is a design move made every time an analog-computing paper writes a cleaner ODE than the one its derivation literally produces. It is rarely called out, because in each individual case the algebra is elementary and the reader who reconstructs it locally sees only the end result. The effect compounds across a paper: a sequence of invisible pass-through eliminations can produce a construction whose correctness proof is shorter than its derivation.

Writing the technique down has two payoffs. First, it gives an unambiguous target for mechanization — a polynomial-helper Gröbner check, or a learned lift. Second, it makes a kind of failure legible. A sparse-regression method that does not see pass-through structure will appear to fail on exactly the constructions where pass-through was used most aggressively. The $\alpha^{\beta}$ experiment above is a miniature example of this phenomenon.

The trick has a colloquial Chinese phrase attached to it — 过而不入, “passing through without entering.” A variable that passes through the construction without entering the executed system. This seems to be the right name.

大音希聲，大象無形。

The greatest sound is beyond hearing; the greatest form is without shape.

— 《道德經》 · Daodejing, chapter 41