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MiNO: Cotangent-bundle propagator learning for PDEs

Gnankan Landry Regis N'guessan, Bum Jun Kim

arXiv:2608.15187v1cs.LGcs.CEmath.NA

TL;DR

Scientific machine learning often learns PDE solution fields or solution maps, despite propagators potentially being smoother phase–amplitude objects that govern nonsmooth fronts. MiNO learns propagators on the cotangent bundle and, on discontinuous advection, reaches the finite-window accuracy limit while a balanced PINN remains near its initial error.

  • Problem

    PDE learning commonly targets nonsmooth solution fields or solution maps, while the potentially smoother propagator governing sharp propagation remains a comparatively underused target.

  • Method

    MiNO learns cotangent-bundle phase–amplitude data by enforcing eikonal and transport equations, then reconstructs solutions through oscillatory integration.

  • Results

    On matched-budget discontinuous advection, MiNO reaches its finite-reconstruction-window accuracy limit within 10,000 steps, while an NTK-balanced PINN remains near its initial error.

  • Takeaways & Limitations

    Residuals can certify the learned canonical relation and separate generator error from the frequency-truncation tail, extending evaluation beyond reconstructed-field accuracy.

  • Takeaways & Limitations

    Reconstruction accuracy depends on the window size and omitted frequency tail, so the window or node count must change when the datum spectrum exceeds it.

Abstract

from arXiv · show

Scientific machine learning for partial differential equations commonly targets solution fields, as in physics-informed neural networks, or solution maps, as in neural operators. We study a third target: the propagator itself, a phase and amplitude in phase space. The motivation is a gap in regularity. A transported discontinuity is nonsmooth in space and time, yet the rule that moves it can be a polynomial phase carrying unit amplitude, so the object that generates an evolution can be far smoother than the field it generates. The microlocal neural operator (MiNO) learns that object, using the eikonal equation for the phase and the transport equation for the amplitude, and recovers the solution by an oscillatory integral. Sharp fronts and caustics then belong to propagation geometry rather than to a field fitted pointwise. Small residuals certify more than the reconstructed field. They place the learned canonical relation, the geometry that carries singularities, close to the exact one, and they separate trainable error from the frequency-truncation tail. On a matched-budget discontinuous-advection benchmark, MiNO stops improving within 10,000 steps at the accuracy limit of its finite reconstruction window, a limit predicted in closed form, whereas a physics-informed neural network with neural-tangent-kernel loss balancing stays near its initial error. On smooth advection, the mean error is $3.84\times10^{-3}$ for MiNO and $3.12\times10^{-2}$ for a supervised Fourier neural operator. Single-branch MiNO is the smallest model compared, and one trained generator serves five unseen initial conditions without retraining.

1. Introduction

MiNO shifts PDE learning from fields or solution maps to a phase–amplitude propagator in phase space, reconstructing solutions through oscillatory integrals. This representation targets smooth propagation geometry behind nonsmooth fronts and improves matched-budget discontinuous-advection performance while enabling reuse across initial conditions.

  • Propagator learning: MiNO learns a phase–amplitude Fourier-integral representation of the propagator on the cotangent bundle, enforcing eikonal and transport equations before oscillatory reconstruction.The phase describes propagation, while the amplitude describes strength.
  • Motivation: A transported discontinuity is nonsmooth in space–time, whereas its propagator can have a polynomial phase and unit amplitude in phase space.For unit-speed advection, the square wave is in L2(R) but in no H^s(R) with s≥1/2, while its propagator has phase xξ−tξ.
  • Results: Within 10,000 optimization steps, MiNO reaches the closed-form accuracy limit of its finite reconstruction window, while a PINN with NTK loss balancing remains near its initial error.The benchmark uses matched budgets and discontinuous advection.
  • Results: MiNO satisfies the eikonal Cauchy datum by construction, eliminating the initial-condition penalty that PINNs must balance against equation residuals.Both methods receive the same governing equation but learn different objects.
  • Generalization: One trained generator serves five distinct initial conditions without retraining because new data enter only through Fourier transforms during reconstruction.This reuse follows from the linearity of the representation rather than training-set coverage.
  • Contributions: The contributions establish propagator learning and MiNO through chartwise stability theory and one- and two-dimensional propagation experiments.The paper argues that neural PDE difficulty depends on the learned object as well as the governing equation.

2. Representational targets and propagator formulation

This section distinguishes field, operator, and propagator learning by their learned objects and ambient spaces, then formulates propagator learning as chartwise phase–amplitude data on the cotangent bundle. The phase generates bicharacteristic geometry through a localized Fourier-integral representation, unlike a learned pseudodifferential symbol, whose canonical relation remains diagonal.

  • 2.1. Solution-field learning: PINNs learn a solution field u_θ(t, x) on the base space–time manifold, using equation residuals and training terms without changing the learned object’s domain.The field remains a function on [0, T] × Ω.
  • 2.2. Solution-operator learning: Neural operators learn the input–output map from initial conditions to solution functions, typically through integral kernels and pointwise nonlinearities trained on paired data.Their learned object is a map between function spaces rather than a field or propagator geometry.
  • 2.3. Propagator learning: Propagator learning targets chartwise phase–amplitude data on the time-parameterized cotangent bundle, generating evolution through an oscillatory integral.The phase generates the bicharacteristic canonical relation in the admissible homogeneous hyperbolic setting.
  • 2.3. Propagator learning: The propagator formulation considers linear Cauchy evolutions on Euclidean, periodic, or localized domains, with sign and quantization conventions affecting the eikonal and amplitude equations.Operators sharing a principal symbol can have different amplitude equations because of quantization and subprincipal-symbol choices.
  • 2.4. Comparison of the three targets: The three targets occupy distinct ambient spaces: fields over space–time, maps between function spaces, and phase–amplitude data that make canonical geometry explicit.Table 1 compares their learned objects, ambient spaces, training signals, and outputs.
  • 2.5. Bicharacteristic kernel relation and locality: In a localized homogeneous hyperbolic FIO chart, an admissible homogeneous phase and smooth amplitude define a kernel relation given by the Hamiltonian flow.The proposition is local, while global propagation requires admissible branches, cutoffs, and branch or Maslov transition data.
  • 2.5. Bicharacteristic kernel relation and locality: Caustics arise from degeneracies of the base projection of a propagated solution Lagrangian, distinct from mixed-Hessian degeneracy in the chosen phase chart.The Airy generating-family normal form separates these two geometric nondegeneracy conditions.
  • 2.5. Bicharacteristic kernel relation and locality: A learned pseudodifferential symbol cannot transport singularities because its fixed identity phase yields the diagonal canonical relation and preserves the wavefront set.This limitation is structural rather than a consequence of insufficient network capacity.

3. Microlocal neural operators

MiNO learns a phase-space propagator by enforcing eikonal and transport structure on the generator rather than fitting the solution field directly. Its construction supports error control for both reconstructed solutions and propagation geometry, while one trained generator transfers across initial conditions under a fixed symbol.

  • Amplitude specialization: All trained experiments use the constant-amplitude specialization A* = 1 with psub = 0, matching chart and experimental residuals at the exact zero target.For scalar advection, the experiments use ∂tA + c∂xA = 0 rather than the half-density equation.
  • Propagator learning: MiNO learns the propagator generator on T*Ω, using eikonal and transport residuals instead of equations posed directly on Ω for the field.The objective targets the phase and amplitude that generate the evolution through an oscillatory reconstruction.
  • Error control: Continuous phase and amplitude residuals yield estimates for reconstruction error, output-operator error, and canonical-geometry error through the paper’s theorems and corollaries.These conversions are stated for continuous residual norms and return both solution and propagation-geometry guarantees.
  • Phase and amplitude parameterization: Time-factored phase parameterization enforces Φθ(0, x, ξ) = x·ξ identically, making the eikonal Cauchy datum exact and its initial-condition loss zero.The network parameterizes the full time-averaged phase increment.
  • Operator-like reuse across initial conditions: A generator trained for one symbol applies to different initial conditions without retraining, because initial data enter only through û0 and the propagator is linear in û0.This operator-like reuse differs from amortized operator learning from input–output pairs.

4. Theory of propagator learning

The theory links eikonal and transport residuals to phase–amplitude reconstruction while separately certifying the learned canonical relation. It also separates continuous residual, quadrature, and frequency-truncation errors, including the sharp obstruction for discontinuous data.

  • Canonical-graph control: Differentiated eikonal residuals bound the distance between the learned canonical relation and the exact propagation geometry.The scalar residual controls phase value, whereas its x- and ξ-derivatives control the generated canonical graph.
  • Phase and reconstruction stability: Hamilton–Jacobi residuals control phase error, and compact-chart reconstruction stability combines that phase contribution with a separate amplitude discrepancy.These results do not require positive homogeneity or mixed-Hessian nondegeneracy.
  • Transport reconstruction: Scalar-transport residual estimates control complete phase–amplitude propagator reconstruction for exact or approximate Cauchy data.The resulting bounds are datum-uniform at the continuous-residual level.
  • Error decomposition: The full-solution bound separates continuous residual error from the frequency tail, while numerical quadrature contributes an additional error term.This decomposition distinguishes errors from learning, truncation, and numerical integration.
  • Frequency truncation: The Ξ^-1/2 L2 truncation rate for square-wave data is an obstruction caused by the finite reconstruction window, not by the learned geometry.The bound applies to discontinuous data after frequency truncation and corresponds to the L2 form of spectral approximation’s classical obstruction.

5. Experiments

Across smooth, discontinuous, variable-speed, wave, caustic, and microlocal benchmarks, MiNO consistently reconstructs advection propagators accurately, while PINN performance can stagnate or collapse. Its finite frequency window explains the square-wave error floor, and one trained generator transfers across distinct initial conditions without retraining.

  • Smooth advection: 3.84 × 10−3 ± 2.18 × 10−3 is MiNO’s error versus 3.12 × 10−2 ± 4.27 × 10−3 for supervised FNO on smooth advection.At 200,000 steps, MiNO reaches 1.32 × 10−3 ± 4.36 × 10−4, while NTK-balanced and causal PINNs reach 5.48 × 10−3 ± 2.04 × 10−3 and 6.09 × 10−1 ± 2.07 × 10−1.
  • Discontinuous advection: 0.103 ± 0.001 is MiNO’s 50,000-step square-wave error, after reaching 0.105 ± 0.001 at 10,000 steps.The exact relative L2 truncation tail is 0.1028 for Ξ = 100 and datum half-width a=0.3, with leading asymptotic value 0.1030.
  • Discontinuous advection: 0.943 ± 0.011 is the PINN’s 500-step square-wave error, compared with MiNO’s 0.299 ± 0.154; at 50,000 steps, their errors are 0.950 ± 0.003 and 0.103 ± 0.001.The log–log slopes are −0.21 for MiNO and +0.001 for the PINN, while the error ratio grows from 3.15× to 9.19×.
  • Variable-speed advection: 0.052 ± 0.043 and 0.112 ± 0.021 are MiNO’s Gaussian and square-wave errors for spatially varying-speed advection, versus 0.801 ± 0.001 and 0.739 ± 0.001 for NTK-balanced PINN.MiNO outperforms both field baselines for both initial data, with error ratios ranging from approximately 2.4 to 15.4.
  • Caustics and microlocal structure: Below 10−3 is MiNO’s relative L2 error at every sampled time through and past the focal time when periodic wrap-around is removed.At finite wavelength, the reconstruction remains finite and smooth at the focal time, while the singularity lies in the base projection of the solution Lagrangian.
  • Transfer and microlocal diagnostics: One generator trained once for a fixed symbol applies to structurally distinct initial conditions without retraining because the initial condition enters only through û0.In the resolved high-frequency band of a transported discontinuity, generator reconstruction follows the exact ridge whereas a low-pass surrogate does not.

6. Discussion and conclusions

The discussion positions propagator learning as a complementary third target that learns phase–amplitude geometry on the cotangent bundle rather than fitting solution fields pointwise. Its evidence supports this target when propagation geometry is simpler than the projected solution, while emphasizing scope limits and separating generator control from truncation error.

  • Regime boundary: Propagator learning is most relevant when propagation geometry is simpler than the projected solution, including transport, waves, sharp fronts, caustics, and inverse propagation.The discussion frames field learning and neural operators as complementary regimes for smooth observable solutions and large trajectory datasets, respectively.
  • Scope and limitations: The evidence is scoped to linear one- and two-dimensional problems with a single phase chart, zero subprincipal symbols, constant target amplitudes, and stated model comparisons.The comparison covers two PINN variants and one supervised FNO at the reported budgets, including an FNO test with a localized Gaussian outside its sinusoidal training distribution.
  • Error control: Residuals control the phase, canonical graph, and truncated propagator while separating generator error from the frequency-truncation tail.For discontinuous data, the tail decays at rate Ξ^-1/2 and sets a reference scale for late-budget error on the chosen reconstruction window.
  • Computational evidence: On discontinuous advection, MiNO reaches the reconstruction-window tail while the NTK-balanced PINN remains near its initial error over the matched sweep.The comparison identifies a distinct optimization regime in the wavefront-dominated benchmark.
  • Computational evidence: MiNO attains lower mean error than the supervised FNO on smooth advection without using labeled trajectories.The reported operating point compares MiNO with a supervised neural operator trained from labeled pairs, whereas MiNO uses the governing equation.

A. Notation

The notation is organized to mirror the paper’s construction, from phase-space geometry through learned reconstruction, residuals, and error estimates.

  • A. Notation: Table A.1 orders recurring symbols by the construction’s progression, covering phase-space coordinates, canonical geometry, reconstruction, residuals, and error estimates.Symbols used only in a single experiment or proof are defined where they occur.

B. Related work · B.1. Solution-field learning and its remedies

The related-work taxonomy contrasts what each method learns, how physics enters, and whether propagation geometry is represented explicitly. For solution-field learning, PINNs and their remedies retain the field as the learned object while addressing discontinuities through weak forms, moving collocation points, or training modifications.

  • B. Related work: The comparison organizes methods by learned object, incorporation of physics, and explicit representation of propagation geometry.This appendix positions MiNO within a three-target taxonomy.
  • B.1. Solution-field learning and its remedies: PINNs learn a field u_θ(t, x) by minimizing strong-form PDE, initial-condition, and boundary residuals at collocation points.Derivatives are computed by automatic differentiation, within a flexible mesh-free framework also used for inverse problems.
  • B.1. Solution-field learning and its remedies: PINNs are effective for smooth fields, but transported discontinuities expose difficulty for pointwise strong-form residual training.The supplied passage introduces this limitation but ends before specifying its full formulation.
  • B.1. Solution-field learning and its remedies: Weak-form training addresses discontinuities by replacing an undefined shock residual with a Kružkov entropy formulation.This remedy changes the training formulation without changing the learned object.
  • B.1. Solution-field learning and its remedies: Lagrangian PINNs move collocation points along characteristics, yielding a frame with a more rapidly decaying n-width for convection-dominated solution manifolds.This is another remedy that retains solution-field learning while adapting the collocation frame.
  • B.1. Solution-field learning and its remedies: Additional remedies include causal weighting, gradient or NTK loss balancing, and modified MLPs with Fourier features.The passage lists these approaches as alternatives that address the same difficulty without changing the learned object.

B.2. Solution-operator learning · B.3. Learned eikonal and analytic phase-space methods

The paper contrasts neural operators that learn solution maps with MiNO, which explicitly parameterizes propagation geometry through a reusable phase–amplitude generator. It also distinguishes MiNO from learned first-arrival eikonal solvers and analytic high-frequency methods because MiNO targets PDE propagator data while accommodating branch structure.

  • B.2. Solution-operator learning: FNO and DeepONet learn input–output maps between function spaces, with FNO combining truncated spectral convolutions and pointwise operations.Both approaches have approximation theory with explicit error bounds.
  • B.2. Solution-operator learning: MiNO explicitly parameterizes propagation geometry through a generator, whereas standard neural operators encode it implicitly through the learned map and training distribution.The solution map is synthesized from the generator.
  • B.2. Solution-operator learning: The generator can be reused across initial conditions because its representation is linear in u0, unlike operator networks trained as amortized maps from pairs.The supervised FNO comparison illustrates this distinction at one smooth-advection operating point.
  • B.3. Learned eikonal and analytic phase-space methods: EikoNet, PINNeik, and related tomography methods learn single-valued first-arrival eikonal solutions, including possibly nonsmooth viscosity solutions.Fast-marching and sweeping methods address the same first-arrival problem on grids.
  • B.3. Learned eikonal and analytic phase-space methods: Phase-space, level-set, and ray-tube methods lift or decompose multivalued arrival structures, while MiNO targets PDE propagator phase–amplitude data and accommodates branch structure.Architectures using Hamilton–Jacobi representation formulas target the equation’s solution.
  • B.3. Learned eikonal and analytic phase-space methods: Maslov, Gaussian-beam, frozen-Gaussian, and curvelet representations prescribe phase-space structure that remains useful at focal points.Fast and butterfly FIO algorithms apply known oscillatory kernels through hierarchical low-rank factorization.

B.4. Learned symbols and microlocal neural architectures … C.1. Class 1: first-order transport 𝑝(𝑥, 𝜉) = 𝑐(𝑥)𝜉, 𝑑= 1

The sections distinguish MiNO’s moving canonical relation from symbol-learning approaches with a fixed identity phase, then derive its phase–amplitude representation for scalar transport. For first-order advection, the eikonal phase follows characteristics while the amplitude satisfies the corresponding scalar transport equation under the stated normalization.

  • B.4. Learned symbols and microlocal neural architectures: Phase-space learning for PDE propagators differs from finite-dimensional dynamics, where learned objects include Hamiltonians or symplectic maps.Prior PDE propagator-learning targets remained fields on the base manifold or maps between function spaces.
  • B.4. Learned symbols and microlocal neural architectures: MiNO learns the admissible phase of a homogeneous hyperbolic FIO, unlike pseudo-differential symbol models whose fixed identity phase cannot move wavefront sets.The symbol-learning neighbor retains the solution operator as its target, whereas MiNO learns a canonical relation locally represented by the graph of χ_t.
  • B.4. Learned symbols and microlocal neural architectures: Microlocal learning also informs imaging through FIO-inspired kernels and canonical-relation-guided wavefront extraction, but those methods target imaging maps or singularity sets.The cited imaging approaches use FIO priors for variable-background wave imaging or known Radon-transform geometry for limited-angle reconstruction.
  • B.5. Classical microlocal theory and the present contribution: MiNO represents classical geometric-optics structure, including FIOs, eikonal–transport hierarchies, canonical relations, Maslov transitions, and uniform analysis at degenerate projections.The passage locates these ingredients in established microlocal theory and geometric optics.
  • C. Transport residuals for scalar advection and quadratic multipliers: The initial Fourier label ξ remains fixed, while the current covector is ζ := ∇_xΦ(t, x, ξ), and A denotes a pulled-back chart amplitude.The section distinguishes this fixed-label parameterization from the current covector variable.
  • C. Transport residuals for scalar advection and quadratic multipliers: The full Hamiltonian derivative governs a free phase-space amplitude, whereas a pulled-back chart amplitude follows transport derived from its generating function.The ansatz describes one normalized dynamical branch, with localization cutoffs and explicit external multibranch weights suppressed.
  • C. Transport residuals for scalar advection and quadratic multipliers: The propagator reconstruction uses ∫ℝ^d e^iΦ(t,x,ξ) A(t,x,ξ) û_0(ξ) dξ with Φ(0,x,ξ) = x·ξ and A(0,x,ξ) = 1.These initial conditions specify the oscillatory-integral ansatz used for the evolution.
  • C.1. Class 1: first-order transport 𝑝(𝑥, 𝜉) = 𝑐(𝑥)𝜉, 𝑑= 1: For ∂_tu + c(x)∂_xu = 0, the eikonal equation follows characteristics ẋ = c(x), while the amplitude satisfies scalar advection; the resulting solution is u_0(X(0;t,x)).Along characteristics, Φ is constant and its covector obeys d(∂_xΦ)/dt = −c′(x)∂_xΦ; for c(x)=x, Φ=e^-txξ and Δ_xΦ=0.

C.2. Class 2: quadratic Fourier multiplier 𝑝(𝜉) · C.3. Extension to other symbols

For the x-independent quadratic multiplier, the exact propagator has phase Φ(t,x,ξ)=x·ξ−tp(ξ) and unit amplitude, satisfying the eikonal and amplitude equations exactly. Extensions retain the invariant geometric transport term while requiring subprincipal and higher-order corrections for general symbols and finite-frequency evolution.

  • C.2. Class 2: quadratic Fourier multiplier p(ξ): The exact propagator for the x-independent multiplier p(ξ)=|ξ|^2/2 is the Fourier multiplier e^−itp(ξ).Equation (C.1) is exact for this free Schrödinger case.
  • C.2. Class 2: quadratic Fourier multiplier p(ξ): The corresponding phase is Φ(t,x,ξ)=x·ξ−tp(ξ), with amplitude A≡1.This phase-amplitude pair represents the exact quadratic Fourier multiplier propagator.
  • C.2. Class 2: quadratic Fourier multiplier p(ξ): The eikonal residual vanishes exactly because ∇xΦ=ξ and ∂tΦ+p(∇xΦ)=−p(ξ)+p(ξ)=0.The quadratic multiplier therefore satisfies the phase equation without approximation.
  • C.2. Class 2: quadratic Fourier multiplier p(ξ): Since ∇²xΦ=0 and all amplitude derivatives vanish, both leading transport and exact finite-wavelength amplitude equations are satisfied.This exactness follows from the linear-in-x phase and constant amplitude.
  • C.3. Extension to other symbols: For p(ζ)=ζᵀHζ/2+ℓ·ζ+c0, the leading half-density transport operator includes the invariant geometric divergence term and, when applicable, ip_subA.The quadratic symbol is parameterized by symmetric H=Hᵀ, while lower-order quantization effects enter through the subprincipal symbol.
  • C.3. Extension to other symbols: For the considered quadratic multiplier, p_sub=0, and the exact unit-scale Schrödinger amplitude equation sets the leading transport operator to i tr(H∇²xA)/2.In semiclassical scaling, the same correction is O(ε).
  • C.3. Extension to other symbols: For arbitrary x-dependent learned amplitudes, the stated transport residual is only the leading WKB equation, while exact finite-frequency evolution includes higher-order terms.A nonzero lower-order Weyl symbol also contributes its own unit-scale action on the amplitude.
  • C.3. Extension to other symbols: The extension discussion applies the same framework to additional charts, including an x-dependent quadratic lens and the two-dimensional half-wave case.The direct Class 2 reduction uses ∇xp=0 and A≡1.

D. Proofs … E. Experimental design and implementation details

The appendices prove that the learned phase generates the bicharacteristic flow and establish stability bounds linking residuals in phase and amplitude to reconstruction and canonical-geometry errors. They also derive scalar-transport and frequency-tail estimates, while the experimental-design section provides an evaluation overview.

  • D. Proofs: The appendix collects proofs for the statements in Sections 2 and 4 under the preceding setting’s notation and hypotheses.
  • D.1. Bicharacteristic kernel relation: The kernel wavefront relation is contained in the graph of the time-t bicharacteristic flow χ_t generated by the Hamiltonian vector field H_p.The eikonal phase with initial datum ϕ(0, x, ξ) = x⋅ξ identifies characteristics with bicharacteristics.
  • D.2. Hamilton–Jacobi phase and reconstruction stability: Hamilton–Jacobi residuals satisfy a linear characteristic equation, and flow estimates yield reconstruction-stability bounds for phase and amplitude errors.The proof uses the fundamental theorem of calculus, characteristic integration, Liouville’s formula, and Minkowski’s inequality.
  • D.3. Canonical-geometry estimates: Phase residual derivatives control both covector and footpoint deviations from the exact canonical geometry through Lipschitz estimates and Gronwall’s inequality.The mixed-Hessian assumption makes the generated relation a local canonical graph, but is not needed for the lemma’s differential inequalities.
  • D.4. Scalar-transport stability and frequency-tail estimates: Scalar-transport phase and amplitude errors obey transport equations driven by their residuals, and operator reconstruction errors follow from Hilbert–Schmidt bounds.The argument combines characteristic integration, Liouville’s formula, Minkowski’s inequality, and the real-phase estimate.
  • D.4. Scalar-transport stability and frequency-tail estimates: Frequency-tail estimates are obtained using Plancherel’s theorem, sin^2(aξ) ≤ 1, changes of variables, and integration by parts.These steps establish the cited truncation estimates for the exact and learned operators.
  • E. Experimental design and implementation details: Table E.1 presents an evaluation overview for the experimental-design and implementation-details section.

E.1. Evaluation design … E.4. Baseline configurations

The evaluation combines propagation, reconstruction, reuse, inverse, dimensional, efficiency, and supervision studies under a defined protocol. MiNO experiments use specialized phase-space residuals, fixed network and reconstruction choices, and comparisons with configured PINN, causal, and FNO baselines.

  • E.1. Evaluation design: The evaluation covers propagation regimes, representational fidelity and reuse, inverse recovery, dimensional extension, efficiency, and supervision.Table E.1 associates each study with its objective and benchmark.
  • E.1. Evaluation design: Representation diagnostics use exact generators, while learning benchmarks use learned-generator specialization and report errors at final time or over the full space–time grid.Seed and per-seed aggregation conventions are documented separately.
  • E.1. Evaluation design: Cross-paradigm baselines train over five seeds for 200,000 steps, while the supervised FNO uses labeled solution data under its own epoch budget; all runs use one NVIDIA A100 GPU.The field-learning baselines are the NTK-balanced PINN and causal PINN.
  • E.2. Experimental residual specialization: MiNO parameterizes phase as x·ξ+t h_θ and amplitude as 1+t γ_θ, using composite trapezoidal reconstruction by default and Filon reconstruction for point-focus diagnostics.The phase increment is directly parameterized in the experiments.
  • E.2. Experimental residual specialization: The experimental residual specializes the full Hamiltonian derivative to constant amplitude A*=1 and p_sub=0, with symbol derivatives evaluated at the learned covector.The general chart-amplitude formulation is not identical away from this target, but both formulations share the same exact zero in the trained subclass.
  • E.3. Networks, optimization, and reconstruction: The phase network has four width-64 hidden layers and 32 sinusoidal features, while the amplitude network has three width-64 hidden layers and 16 sinusoidal features.The phase features use fixed scales 0.25 in x and 0.1 in ξ; the experimental amplitude enforces its initial value.
  • E.3. Networks, optimization, and reconstruction: Trained MiNO and baseline results generally report five-seed means and sample standard deviations, while the multi-initial-condition generator uses one seed and inverse experiments use five noise seeds per configuration.The inverse study totals sixty configurations across noise levels.

F. Limitations … Funding

The paper reports a scoped evaluation with narrower trained configurations than the full construction, alongside authorship, conflict-of-interest, availability, and funding statements. The implementation and reproducing experiment records are available from the corresponding author on request, and no specific grant supported the research.

  • F. Limitations: The evaluation covers linear propagation problems in one and two space dimensions and compares NTK-balanced and causal PINNs with a supervised FNO.It does not span every benchmark dataset or current field-learning and operator-learning architecture.
  • F. Limitations: Every trained run uses a single phase chart with m_ν = 1, so chart crossing and nontrivial Maslov transitions are not exercised.The two-branch wave configuration superposes two single-chart generators under explicit external weights rather than transitioning between charts.
  • F. Limitations: All trained runs use p_sub = 0 and the constant-amplitude specialization A* = 1.These settings are narrower than the construction described in Section 3.
  • CRediT authorship contribution statement: Gnankan Landry Regis N’guessan contributed to conceptualization, methodology, software, analysis, investigation, data curation, validation, visualization, and writing.Bum Jun Kim contributed to formal analysis, visualization, supervision, validation, and writing.
  • Declaration of competing interest: The authors declare no known competing financial interests or personal relationships that could have influenced the reported work.
  • Data availability: The MiNO implementation and experiment records reproducing the reported numbers are available from the corresponding author on request.
  • Funding: The research received no specific grant from any public, commercial, or not-for-profit funding agency.
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