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Tractable Inference with Kolmogorov-Arnold Networks

Juan Diego Toscano, Sascha Ranftl, Zhiwei Gao, Li-Lian Wang, George Em Karniadakis

arXiv:2610.29909v1physics.comp-phcs.LGmath.NA

TL;DR

Conventional neural networks often estimate surrogate expectations and other integral quantities by sampling or multidimensional quadrature, which becomes costly in high dimensions or repeated evaluations. TOUCKANs use polynomial outer functions to make moments tractable under factorizing inputs. Their deterministic moments compare favorably with fixed-budget Monte Carlo errors across benchmarks and support integral constraints in physics applications.

  • Problem

    Conventional neural networks, including KANs, evaluate expectations and other surrogate quantities through sampling or multidimensional quadrature, costly in high dimensions or repeated evaluations.

    Source: §1 · p. 2, §2 · p. 4

  • Method

    TOUCKANs use finite-degree polynomial outer functions to form decomposable sum–product representations, reducing moments under factorizing inputs to deterministic recursions over one-dimensional leaf moments.

    Source: §1 · p. 2, §2 · p. 5

  • Results

    Across uniform and Gaussian benchmarks through D = 50, deterministic mean and variance errors stayed below median fixed-budget Monte Carlo errors in every case; TOUCKANs also conserved Cahn–Hilliard mass to machine precision and supported four Schrödinger eigenpairs in D = 2, 3, and 5.

    Source: §1 · p. 3, §5 · p. 22

  • Takeaways & Limitations

    Repeated moments can be computed without new true-function evaluations, including from fixed observations or archived simulation data.

    Source: §5 · p. 22

    The formulation assumes factorizing input measures, with correlated inputs identified as a primary extension.

    Source: §5 · p. 23

Abstract

from arXiv · show

Machine-learning surrogates in computational science often require not only accurate predictions but also expectations, variances, conserved integrals, and normalization constants. Conventional neural networks, including Kolmogorov--Arnold networks (KANs), generally evaluate these quantities through Monte Carlo sampling or multidimensional quadrature, making repeated inference costly, particularly in high dimensions. We introduce KANs with Tractable Operators for Uncertainty and Constraints (TOUCKANs), whose finite-degree polynomial outer functions yield a decomposable sum--product representation. Under factorizing input measures, expectations and higher moments then reduce algebraically to deterministic recursions over one-dimensional leaf integrals. Independent and shared inner-block variants form a parameter-efficiency hierarchy, and a universal-approximation proof for its most restricted member extends to the complete class. The same deterministic operators allow conserved integrals and normalization to be imposed throughout training. Across uniform and Gaussian benchmarks through 50 dimensions, deterministic mean and variance errors remained below the median fixed-budget Monte Carlo errors in every case. TOUCKANs also conserved Cahn--Hilliard mass to machine precision and imposed normalization and orthogonality through deterministic moments, enabling joint computation of the lowest four Schrödinger eigenpairs in two, three, and five dimensions. TOUCKANs therefore connect KAN approximation with tractable probabilistic circuits, making statistical operators and integral constraints directly available within an expressive neural surrogate.

1. Introduction

Scientific surrogates need efficient statistical and integral quantities, but conventional neural networks often rely on costly sampling or multidimensional quadrature. TOUCKANs build tractable deterministic moment operators and integral constraints into KANs.

1. Introduction

  • Finite-degree polynomial outer functions yield decomposable sum–product representations, reducing expectations and variances to deterministic recursions over one-dimensional leaf moments.The construction extends to arbitrary depth and polynomial observables.Source: §1 · pp. 2–3
  • Independent, full-head, diagonal, and rank-one inner parameterizations form a hierarchy of progressively smaller parameter footprints, with universal approximation proved for the restricted rank-one class and extended to the full hierarchy.Shared inner formulations reduce parameters without changing moment recursions.Source: §1 · pp. 2–3
  • Deterministic integral evaluations impose conserved linear functionals and normalization by construction, remaining differentiable during training and valid for every parameter value under the quadrature-discretized functional.Source: §1 · p. 3
  • Across uniform and Gaussian benchmarks through D = 50, deterministic mean and variance errors stayed below median fixed-budget Monte Carlo errors; the framework also conserved Cahn–Hilliard mass to machine precision and computed four Schrödinger eigenpairs in D = 2, 3, and 5.Normalization and orthogonality were imposed through deterministic moments for the eigenpair computations.Source: §1 · p. 3
  • TOUCKANs connect KAN approximation with tractable probabilistic circuits, making statistical operators and integral constraints available during inference and training.Source: §1 · p. 3

2. From Kolmogorov–Arnold superposition to tractable inference

Scientific applications need expectations, moments, variances, and integral constraints in addition to accurate function approximation. This section derives tractable inference by imposing polynomial outer functions on KAN-style compositions.

2. From Kolmogorov–Arnold superposition to tractable inference

  • General neural representations typically evaluate expectations and related quantities by sampling or multidimensional quadrature, which becomes costly at high dimension or when recomputed during training or inference.The observable uses input density ρ(x), while h specifies the quantity of interest.Source: §2 · p. 4
  • The resulting moment reduction is algebraically exact, with remaining approximation arising only from quadrature used to evaluate the one-dimensional leaf moments.Source: §2 · p. 4
  • Finite-degree polynomial outer functions let the multinomial theorem express expanded monomials as products of powers of univariate inner functions, whose expectations separate under factorizing input densities.Recursive expansion extends the result to deeper polynomial compositions and polynomial observables; first and second moments determine variance.Source: §2 · p. 5

3. Method

TOUCKANs use polynomial outer functions and univariate inner branches to create decomposable structure, making moments computable by deterministic one-dimensional recursions. Shared inner-block variants reduce parameters, while the same operators support conserved-integral and normalization constraints.

3.1. A tractable KAN architecture

  • Finite-degree polynomial outer functions over sums of univariate branches yield decomposable structure; shared inner-block variants reduce parameters while preserving universal approximation across the hierarchy.The hierarchy ranges from independent per-dimension networks to rank-one sharing.Source: §3.1 · pp. 5–7

3.2. Exact expectation via binomial moment recursion

  • Under factorizing input measures, expectations reduce to dimensionwise binomial recursions over one-dimensional leaf moments, with quadrature error confined to those leaf integrals.The result is exact for the quadrature-induced discrete product measure; continuum error comes from one-dimensional quadrature.Source: §3.2 · p. 8

3.3. Exact variance via cross-moment recursion

  • Variance requires bivariate cross-moment recursions because distinct combination variables generally depend on the same input coordinates and are therefore dependent.For fixed polynomial degree and quadrature size, the arithmetic cost is proportional to DQ^2; the construction extends to higher moments.Source: §3.3 · p. 9

3.4. Exact enforcement of conserved integrals

  • Weighted one-dimensional leaf integrals let the expectation recursion evaluate conserved linear functionals, which can be enforced by a differentiable output reprojection.The prescribed functional holds for every parameter value under the quadrature-discretized functional.Source: §3.4 · pp. 10–11

3.5. Exact density estimation: normalization as a hard constraint

  • Dividing the squared surrogate by its deterministic second moment enforces unit normalization for every parameter value, provided the second moment is positive.The continuum normalization inherits the accuracy of the one-dimensional quadrature rules.Source: §3.5 · p. 11

4. Results

The results examine parameter efficiency, high-dimensional moment inference, exact mass conservation, and joint Schrödinger eigenpair computation. Deterministic propagation beats median fixed-budget Monte Carlo across the tested benchmark dimensions, while supporting integral constraints in the PDE and eigenvalue experiments.

4.1. Parameter efficiency

  • The diagonal specialization used 11.8 times fewer parameters than the independent variant while matching variance accuracy; rank-one increased relative L2 error to approximately 8%.The diagonal form was selected for the high-dimensional inference experiments; the reported finite-budget loss for rank-one does not contradict the universal-approximation result.Source: §4.1 · p. 12

4.2. Tractable inference

  • Across D = 5–50, TOUCKAN mean errors (10^-10–10^-5) and variance errors (10^-8–10^-4) stayed below median fixed-budget Monte Carlo errors in every case.The comparison spans the three benchmarks; Table 2 also reports that TOUCKAN outperforms at least 98% of mean and 89% of variance Monte Carlo replications at D = 50.Source: Table 2 · p. 13, §4.2 · p. 13
  • One TOUCKAN evaluation was up to 2,600 times faster than averaging 200 Monte Carlo runs, matching or improving accuracy except for Bratley at D = 50.The timings are for a warm compiled-recursion evaluation and exclude surrogate training and one-time JIT compilation.Source: Figure 3 · p. 15, §4.2 · p. 15

4.3. Exact mass conservation: Cahn–Hilliard equation

  • All three TOUCKAN variants conserved mass to machine precision; independent TOUCKAN had the lowest relative L2 error (1.2 × 10−4), while the unconstrained MLP had the lowest L∞ error (5.2 × 10−4).A constrained TNN also conserved mass to machine precision, but its relative L2 error was 4.1 × 10−3.Source: §4.3 · pp. 17–19

4.4. Exact normalization: the Schrödinger eigenvalue problem

  • For Gram-matrix evaluation, cross-moment recursion reached reference accuracy with 256, 384, and 640 univariate evaluations in D = 2, 3, and 5, with full evaluations taking 1–3 ms.The D = 5 comparison is limited by the convergence of its 485-node reference quadrature.Source: Figure 5 · p. 18, §4.4 · p. 20
  • Mean relative eigenvalue errors for the jointly computed lowest four eigenpairs were 1.7 × 10−8, 1.4 × 10−7, and 6.1 × 10−5 in D = 2, 3, and 5, respectively.All states were optimized jointly from the first iteration, without an eigenvalue scan or sequential deflation.Source: §4.4 · pp. 19–20

5. Summary

TOUCKANs make statistical moments tractable through a decomposable polynomial structure, with approximation capacity and parameter cost controlled by the inner-block hierarchy. The summary also identifies training-coverage and factorizing-input assumptions as practical boundaries.

5. Summary

  • Rank-one, the most restricted inner parameterization, provides the universality proof that extends to the complete TOUCKAN hierarchy.Independent and shared inner networks offer progressively smaller parameter footprints; parameterization controls the finite-budget accuracy–cost trade-off.Source: §5 · p. 22
  • Across uniform and Gaussian benchmarks through D = 50, deterministic mean and variance errors stayed below median fixed-budget Monte Carlo errors in every case.At D = 50, variance propagation hit a floating-point roundoff floor; for lower-dimensional physics-informed tasks, richer couplings attained the lowest field errors.Source: §5 · p. 22
  • Deterministic moments let TOUCKAN enforce integral constraints during training, conserving Cahn–Hilliard mass to machine precision and enabling joint computation of the lowest four Schrödinger eigenpairs in D = 2, 3, and 5.The Schrödinger setup imposes normalization and orthogonality through deterministic moments.Source: §5 · p. 22
  • Training coverage can become the practical bottleneck: richer joint structure may require substantially larger pointwise training sets than the smooth targets tested through D = 50.Unlike tensorized or separable architectures, TOUCKAN training evaluates complete coordinate tuples rather than an axis-wise Cartesian grid.Source: §5 · pp. 22–23

Appendix A.1. Kolmogorov–Arnold Networks

  • KKANs return to KAN’s two-block structure, approximating both inner and outer functions with smooth learnable components rather than using the exact, highly non-smooth constructive representation.The KART represents continuous multivariate functions through superpositions of univariate functions, while KANs place learnable univariate functions on graph edges.Source: Appendix A.1 · p. 29

Appendix A.2. Sum–Product Networks and Tractable Inference

  • Sum–Product Networks recursively combine sums and products over structured scopes; polynomial outer nonlinearities give KANs a related decomposable multiplicative structure.Decomposability requires distinct children of each product node to have disjoint scopes.Source: Appendix A.2 · pp. 29–30

Appendix B.1. Building intuition with shallow minimal-order networks

  • For quadratic outer activations under factorizing measures, the shallow expansion yields a decomposable representation whose high-dimensional integral reduces to algebraic terms assembled from one-dimensional leaf moments.The stated count is (2D+1) algebraic terms from 2D unique one-dimensional leaf moments per branch.Source: Appendix B.1 · p. 31

Appendix B.2. Higher orders

  • Higher-degree polynomial activations expand into products over more univariate leaves, and the same multinomial reduction extends across layers; numerical leaf integration makes inference near-exact rather than exact.For cubic activations, products involve triples of inner functions; repeated expansion handles trailing orders and deeper layers.Source: Appendix B.2 · pp. 31–32, Appendix B.3 · p. 32

Appendix B.4. Implementation

Under factorizing priors, TOUCKAN moment evaluation can be implemented by replacing leaves with their required power moments and passing the resulting values through the circuit.

Appendix B.4. Implementation

  • Under factorizing priors, the expectation of a leaf-function composition can be computed by replacing each leaf with expectations of its required powers, then evaluating the circuit forward.TOUCKAN additionally tracks powers through the binomial recursion.Source: Appendix B.4 · p. 33

Appendix B.5. Computational complexity scaling

Moment propagation replaces high-dimensional integration with low-dimensional leaf moments and algebraic operations, with computational cost governed by the expanded decomposable representation. Its general result assumes polynomial outer activations, integrable leaf powers, and a factorizing input density.

Appendix B.5. Computational complexity scaling

  • For deep TOUCKANs, tractability is measured against the size of the expanded decomposable computational graph generated by recursive multinomial expansion.The inference procedure replaces high-dimensional integration with low-dimensional moments and algebraic operations.Source: Appendix B.5 · pp. 33–34
  • For fixed polynomial degree, shallow TOUCKAN moment propagation has polynomial cost in input dimension, while the degree of that polynomial grows exponentially with depth.Uncontrolled increases in both depth and polynomial order can cause rapid combinatorial growth.Source: Appendix B.5 · pp. 33–34

Appendix C.1. Expanded decomposable representation

  • Recursive multinomial expansion represents the network as a finite sum of products whose factors each depend on one input variable.The representation uses coefficients and nonnegative integer exponents for the univariate factors.Source: Appendix C.1 · p. 34

Appendix C.2. Moment propagation operator

  • With finite-degree polynomial activations, integrable leaf powers, and a factorizing input density, every statistical moment reduces to one-dimensional leaf moments combined through a finite algebraic composition.For polynomial observables, the result is followed by a forward pass through the expanded decomposable graph.Source: Appendix C.2 · p. 35

Appendix C.4. Proof

The proof uses induction over the computational graph: polynomial activations preserve decomposability, and factorized densities make each resulting monomial integral a product of one-dimensional moments.

Appendix C.4. Proof

  • At the base case, leaves depend on single variables, so their moments are one-dimensional integrals.Source: Proof · p. 35
  • Applying the multinomial theorem to powers of sums of decomposable monomials yields another finite sum of products of univariate functions, preserving decomposability at each layer.Source: Proof · pp. 35–36
  • Because the density factorizes, each monomial term in the integral factors into one-dimensional moments.Linearity of integration supports the decomposition.Source: Proof · p. 36

Appendix D. Universal approximation of TOUCKANs

The most restricted rank-one TOUCKAN class is dense in the continuous functions on [0, 1]^D, so the full parameterization hierarchy is universally approximating. The proof uses shared inner profiles and polynomial outer approximations; its scope is continuous functions.

Appendix D. Universal approximation of TOUCKANs

  • The proof establishes density using a rank-one subclass with shared profiles and dimension-specific scalars, anchored in Laczkovich’s shared-profile superposition theorem [34].The architecture approximates the inner profiles and then the outer function with a Chebyshev polynomial.Source: Appendix D · pp. 36–38
  • All four TOUCKAN parameterizations are dense in C([0, 1]^D) because the rank-one class is dense and contained in each richer class.Their differences concern finite-parameter accuracy and efficiency, not asymptotic expressivity.Source: Appendix D · p. 37
  • The sufficient branch count is asymptotically about 3.41D, and the authors note that this bound is probably not sharp.Experimental branch counts do not test the sharpness of the sufficient bound.Source: Appendix D · pp. 38–39
  • The universality result covers continuous functions, a narrower scope than settings with discontinuous decision boundaries.The paper identifies this as a general limitation of Kolmogorov–Arnold representations, not a limitation unique to TOUCKANs.Source: Appendix D · p. 39

Appendix E. Moment-propagation algorithms

Moment propagation reduces high-dimensional expectations and variances to one-dimensional leaf integrals followed by dimension-wise recursions. The propagation is algebraically exact in those integrals, conditional on factorizing input densities; quadrature is the sole discretization.

Appendix E. Moment-propagation algorithms

  • Under a factorizing input density, first and higher moments are assembled from one-dimensional leaf moments by dimension-wise binomial recursions.The first-moment recursion carries P + 1 partial-sum moments and costs O(QDP^2).Source: Appendix E · pp. 39, 41
  • Only the one-dimensional leaf integrals are discretized by measure-matched Gaussian quadrature; downstream propagation is algebraically exact for the computed moments.The appendix uses Gauss–Legendre rules for uniform marginals and Gauss–Hermite rules for Gaussian marginals.Source: Appendix E · pp. 39–40
  • Centering branches at the origin removes a shared offset that otherwise disrupts high-dimensional computation, and centered evaluation reproduces variance to relative error of order 10^-8 through D = 50.The reported errors at D = 5, 12, 25, and 50 are 1.1 × 10^-8, 9.3 × 10^-9, 3.8 × 10^-8, and 1.7 × 10^-8.Source: Appendix E · pp. 40–41

Appendix E.2. Exact second moment and variance

  • The second-moment recursion tracks joint moments of branch pairs because combination variables share input coordinates and are dependent.The bivariate recursion costs O(Q^2DP^4), in addition to O(Q^2DP^2M) for joint leaf-moment quadrature.Source: Appendix E.2 · p. 43
  • Variance is obtained by assembling the second moment from shifted cross-moments and subtracting the squared first moment.The derivation uses the bivariate recursion and deterministic shifts by the baseline K.Source: Appendix E.2 · p. 43

Appendix F. Sensitivity analysis

Quadrature refinement shows that the production orders for representative moment calculations lie in their saturated error regimes. A separate refinement test finds that Cahn–Hilliard mass conservation persists independently of the rule used to impose the constraint.

Appendix F. Sensitivity analysis

  • At D = 50, Lüthen mean and variance errors saturate from M ≈ 12 at 1.63 × 10^-10 and 1.66 × 10^-8; Schrödinger Gram error reaches about 10^-8 for M ≥ 48.At production order M = 128, the Schrödinger Gram relative error is 1.7 × 10^-8 in Frobenius norm; Fig. F.7 shows both convergence trends.Source: Appendix F · p. 44, Figure F.7 · p. 45
  • Cahn–Hilliard mass drift falls from 1.0 × 10^-2 at N = 8 to about 5 × 10^-15 by N ≈ 128, independently confirming the continuum mass beyond the enforcement rule.The N = 512 production rule lies in the converged regime; Figure F.8 shows refinement and timewise drift across evaluation rules.Source: Appendix F · pp. 45–46, Figure F.8 · p. 46

Appendix G. Randomized quasi-Monte Carlo comparison

The comparison reports lower TOUCKAN errors than median fixed-budget Monte Carlo in every benchmark case, while randomized quasi-Monte Carlo often narrows the gap. The appendix also documents benchmark structure and clarifies that these methods operate in different computational settings.

Appendix G. Randomized quasi-Monte Carlo comparison

  • At N = 2^18, TOUCKAN has lower errors than median RQMC in most cases, with the stated exceptions across Lüthen, Bratley, and Oakley–O’Hagan means and variances.RQMC substantially reduces sampling errors relative to MC; Table G.5 reports median errors over 200 scramblings for RQMC and 200 runs for MC.Source: Appendix G · pp. 46–47, Table G.5 · p. 47
  • At the matched budget, TOUCKAN has lower errors in 22 of 24 cases, with both Bratley moments at D = 50 as exceptions.Table G.5 covers three benchmarks and dimensions 5, 12, 25, and 50; TOUCKAN evaluates trained-surrogate moments, whereas MC and RQMC integrate the true integrand.Source: Appendix G · p. 47, Table G.5 · p. 47, Table G.6 · p. 48
  • The comparison has different cost boundaries: sampling evaluates the true integrand, while TOUCKAN timing excludes training and measures warm moment evaluations after surrogate construction.The analytical benchmarks have inexpensive integrands, so sampling times do not represent data generation from expensive simulations.Source: Appendix G · p. 48

Appendix H. Problem descriptions

  • The benchmark reference moments are exact because independent inputs and sums and products of univariate terms reduce them to one-dimensional integrals.The integrals are available in closed form or by one-dimensional Gaussian quadrature to machine precision.Source: Appendix H · p. 48

Appendix H.1. Lüthen function (uniform inputs)

  • The Lüthen benchmark uses uniform inputs, and its mean and variance references are computed by separating terms into one-dimensional integrals.The reported function includes dimension-dependent interaction terms; the reference integrals use 200-node Gauss–Legendre quadrature.Source: Appendix H.1 · p. 49

Appendix H.2. Bratley function (uniform inputs)

  • The Bratley benchmark is an alternating sum of nested products whose exact variance is reduced to a finite double sum using independent-coordinate moments.Each nested product’s mean follows from E[x_j] = 1/2.Source: Appendix H.2 · pp. 49–50

Appendix H.3. Oakley–O’Hagan function (Gaussian inputs)

  • The Oakley–O’Hagan benchmark uses standard Gaussian inputs, with its variance derived from main effects and pairwise interactions in an order-two ANOVA decomposition.Its function combines univariate linear, sine, and cosine terms with a quadratic form.Source: Appendix H.3 · p. 50

Appendix I. Implementation details

The appendix specifies precision, timing, quadrature, random-seed, and error-reporting conventions used across the experiments.

Appendix I. Implementation details

  • All experiments use double precision, and reported scalar and field errors are relative errors against their references.Field norms are evaluated on each experiment’s specified test grid or test set.Source: Appendix I · p. 51
  • Moment recursions use 200-node Gauss–Legendre rules for uniform inputs, 128-node Gauss–Hermite for Gaussian inputs, and a 512-node midpoint rule for Cahn–Hilliard.Gauss–Hermite weights are normalized to integrate against the standard normal density.Source: Appendix I · p. 51
  • Random seeds vary by experiment: five seeds for parameter efficiency, seed 111 for function benchmarks, seed 9999 for physics experiments, and NumPy RandomState(7) for fixed problem data.Moment timings are warm compiled-kernel times; one-time JIT compilation is reported separately where relevant.Source: Appendix I · p. 51

Appendix I.1. Parameter-efficiency analysis

The parameter-efficiency study compares four inner-block variants while holding their backbone and outer block fixed; the variants differ in how they form per-dimension branches.

Appendix I.1. Parameter-efficiency analysis

  • All four variants share a backbone and 656 outer parameters, while their inner blocks range from independent per-dimension networks to a rank-one Lorentz form.The alternatives are independent networks, full linear heads, a diagonal specialization, and the rank-one form.Source: Appendix I.1 · pp. 51–52
  • The diagonal specialization uses 41 scalars per dimension, compared with about 2,057 parameters per dimension for independent networks and 861 for full linear heads.The rank-one Lorentz form uses one scalar per dimension.Source: Appendix I.1 · p. 51
  • Each variant is trained from five seeds, with Table 1 reporting mean ± standard deviation of relative L2 test errors on 20,000 independent Latin hypercube points.Analytical moments are cross-checked against a 2 × 10^6-sample Monte Carlo estimate of the same network, agreeing within one Monte Carlo standard error.Source: Appendix I.1 · p. 52

Appendix I.2. High-dimensional benchmarks

The high-dimensional benchmark appendix gives model and training configurations, then describes exact-mass reprojection for Cahn–Hilliard models and its conservation results.

Appendix I.2. High-dimensional benchmarks

  • The high-dimensional benchmarks use diagonal inner blocks across three benchmarks and dimensions 5, 12, 25, and 50, with architecture choices varying by benchmark.At dimension 50, the models have 17,783 parameters for Lüthen, 18,482 for Oakley–O’Hagan, and 25,007 for Bratley.Source: Appendix I.2 · p. 52
  • High-dimensional models are trained with Adam warm-up, SSBroyden polishing, and a final ridge-regularized linear refit of the outer coefficients.Warm moment evaluation takes 0.3–0.6 seconds per model, excluding one-time JIT compilation of 1–38 seconds.Source: Appendix I.2 · pp. 52–53

Appendix I.3. Cahn–Hilliard equation

  • The Cahn–Hilliard reference uses an ETDRK4 solver, while the learned models enforce periodicity through Fourier-feature embeddings and fit the PDE residual and initial condition.The TNN baseline is represented as a separable sum of temporal and spatial factors.Source: Appendix I.3 · pp. 53–54
  • Exact-mass reprojection enforces the 512-node quadrature mass functional, with continuum mass conservation limited by the midpoint rule’s spectral accuracy.The correction is constant in space and costs one evaluation of the spatial branch at the quadrature nodes per loss call.Source: Appendix I.3 · p. 54
  • Constrained continuation reduces Cahn–Hilliard mass drift to machine precision, while unconstrained drift remains at the level of field error.The paired comparison does not isolate reprojection’s effect on field accuracy because constrained polishing also continues optimization.Source: Appendix I.3 · p. 55

Appendix I.4. Schrödinger eigenvalue problem

The Schrödinger experiments test TOUCKANs on coupled oscillators whose dense rotation prevents input-coordinate separation. A shared network uses deterministic moments to impose integral constraints and select the lowest eigenstates, and the formulation learns states simultaneously in multivariate settings.

Appendix I.4. Schrödinger eigenvalue problem

  • The coupled oscillator’s dense orthogonal rotation mixes coordinates, so neither its potential nor eigenfunctions separate in the input variables.The target eigenvalues are non-degenerate in each dimension considered, making per-state errors well defined.Source: Appendix I.4 · p. 55
  • A fixed Gaussian envelope carries decay at infinity and turns state inner products into Gaussian expectations of products of network outputs.The cross-moment recursion computes those expectations.Source: Appendix I.4 · p. 56
  • Hard normalization divides each state by its Gram-matrix norm, so normalization holds for every parameter value without appearing in the loss.This construction also removes the trivial zero-state solution.Source: Appendix I.4 · p. 56
  • The deterministic Gram matrix supplies orthogonality penalties, while minimizing tr(G^-1K) selects the subspace of the Ns lowest eigenstates.The trace term assumes G is positive definite; trained Gram matrices remain well conditioned in the reported runs.Source: Appendix I.4 · pp. 58–59
  • TOUCKAN learns all Ns states simultaneously on non-separable multivariate problems, replacing sampled integral constraints with deterministic Gram-matrix orthogonality and a variational trace term.The formulation retains residual loss and trainable eigenvalues, while the envelope carries decay over R^D.Source: Appendix I.4 · p. 60
  • The appendix includes learned-state visualizations for the coupled two-dimensional and five-dimensional oscillators, with reference, surrogate, and error panels.In the five-dimensional slice, E1 and E2 have similar projected profiles; Table 4’s Gram residuals quantify their full-dimensional distinction.Source: Figure I.9 · p. 61, Figure I.10 · p. 62
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