MiNO: Cotangent-bundle propagator learning for PDEs

Gnankan Landry Regis N'guessan; Bum Jun Kim

cs.LGAug 15, 2026

MiNO: Cotangent-bundle propagator learning for PDEs

Authors: Gnankan Landry Regis N'guessan, Bum Jun Kim

Organizations: Axiom Research Group · Department of Applied Mathematics and Computational Science, The Nelson Mandela African Institution of Science and Technology, 404 Nganana, Kikwe, Arumeru, P.O. Box 447, Arusha, 23311, Tanzania · African Institute for Mathematical Sciences, Research and Innovation Centre, KN 3 Road, Gasharu Cell, Kicukiro Sector, Kicukiro District, Kigali, Rwanda · Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan

Abstract

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.

Figures & tables

Figure 1: Graphical overview of the MiNO paradigm. Conventional field learning must fit a moving discontinuity in physical space, while supervised operator learning is trained on labeled pairs. MiNO instead learns smooth phase–amplitude propagator data in phase space and reconstructs the sharp front. The chart at the right sketches the matched-budget discontinuous-advection benchmark of Sec. 5.1.2 , in which MiNO, using no labeled trajectories, settles near the analytic frequency-truncation tail scale while the NTK-balanced PINN stays near its initial error. The plotted MiNO curve and PINN level follow the measured means of Table 2 on a logarithmic error axis, drawn without numerical axes; that PINN has converged to the trivial solution

u\equiv 0

(Sec. 5.1.2 ).

Figure 2: The three representational targets. Field learning targets a solution field over space–time; operator learning targets a map between function spaces; and propagator learning targets chartwise phase–amplitude data on the time-parameterized cotangent bundle. In the admissible homogeneous hyperbolic setting of Proposition 2.2 , the phase generates the bicharacteristic canonical relation.

Paradigm	Learned object	Space	Training signal	Output
PINN	$u_{\theta}(t,x)$	base manifold	PDE residual	solution field
Neural operator	$\mathcal{G}_{\theta}$	function-space map	supervised pairs	solution map
MiNO	$(\Phi_{\theta},A_{\theta})$	cotangent bundle	eikonal and transport	propagator reconstruction

Table 1: Attributes of the three paradigms: learned object, ambient space, training signal, and output.

Figure 3: Cross-paradigm comparison on the 200,000-step runs, plotted on a log scale, for MiNO, the NTK-balanced PINN, and the causal PINN across smooth advection, discontinuous advection, wave propagation, and the Gaussian and discontinuous initial-data cases for

c(x)=1+0.5\sin(\pi x)

of Sec. 5.1.5 . Those two variable-coefficient groups are distinct from the

c(x)=-x

stationary-point study of Sec. 5.1.4 , which is not plotted here. All plotted errors use their respective final-time protocols. For the wave equation, the comparison includes MiNO and the causal PINN, the two implementations that enforce the zero-initial-velocity condition. The supervised FNO is evaluated separately.

Budget (thousand steps)	MiNO	PINN	PINN-to-MiNO ratio ( $\times$ )
$0.5$	$0.299\pm 0.154$	$0.943\pm 0.011$	$3.15$
$1$	$0.179\pm 0.051$	$0.949\pm 0.012$	$5.31$
$2$	$0.136\pm 0.038$	$0.953\pm 0.014$	$7.03$
$5$	$0.127\pm 0.049$	$0.954\pm 0.014$	$7.53$
$10$	$0.105\pm 0.001$	$0.953\pm 0.011$	$9.09$
$20$	$0.105\pm 0.003$	$0.952\pm 0.007$	$9.09$

Table 2: Discontinuous-advection learning curve for MiNO and the NTK-balanced PINN, evaluated over the full space–time grid.

Figure 4: Point-focus reconstruction from the exact generator, with no network trained, plotted against the periodic spectral reference computed on the box

[-L/2,L/2)

. Widening that box removes the periodic-wrap floor; at

L=32

and

L=48

the reconstruction stays below

10^{-3}

relative

L^{2}

through and past the focal time

t=f

Figure 5: Fixed-resolution wave-packet portrait $\log_{10}\lvert T_{\lambda}u(x_{0},\xi_{0})\rvert^{2}$ of a propagated discontinuity at $\lambda=40$ , on a common color scale: (a) the exact grid values, with the two transported jumps marked; (b) the reconstruction from the exact finite-window generator, with no network trained; and (c) a low-pass surrogate keeping the first 12 Fourier modes. The generator reconstruction preserves the resolved high-frequency ridge attached to those jumps, whereas the surrogate removes that ridge while retaining coarse content.

Paradigm	Method	Parameters	Time per step (ms)
Propagator learning	MiNO (single-branch)	58,658	$2.5$
Propagator learning	MiNO (two-branch wave)	117,316	$5.1$
Field learning	PINN (NTK-balanced)	99,649	$2.0$
Field learning	Causal PINN	202,764	$6.3$
Operator learning	FNO (supervised)	329,281	—

Table 3: Parameter counts and per-step wall-clock time on the 200,000-step runs. Every method listed is trained on residuals alone, with no labeled solution data, except the FNO, which is supervised on solutions under a separate epoch budget and is not timed here. The single-branch MiNO and NTK-balanced PINN rows are timed on smooth advection; the two-branch MiNO and causal PINN rows are timed on the wave problem.

Figure 6: Conceptual failure-mode coordinates based on datum regularity and proximity to a projection singularity. The four situations are qualitative prompts for method selection, not measured regimes or decision boundaries.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Meaning
Phase space and data
$d,\Omega,[0,T]$	Spatial dimension; domain $\mathbb{R}^{d}$ , $\mathbb{T}^{d}$ , or a chart; time interval
$T^{*}\Omega$	Cotangent bundle $\{(x,\xi)\}$ ; the phase space of the generator
$x,\xi$	Base point; initial Fourier (fiber) label of the generator
$\zeta$	Current covector $\zeta=\nabla_{x}\Phi_{\theta}(t,x,\xi)$
$u_{0},u(t,x)$	Initial datum; solution of the Cauchy problem

Appendix

Table A.1: List of notation: the symbols used repeatedly in the paper.

Objective	Benchmark	Section
Propagation regimes
Closed-form propagation calibration	smooth advection	Sec. 5.1.1
Sharp-transport regime comparison	discontinuous advection	Sec. 5.1.2
Two-branch propagation	wave equation	Sec. 5.1.3
Inhomogeneous eikonal construction	stationary-point transport	Sec. 5.1.4
Cross-paradigm variable-coefficient comparison	sinusoidal-speed transport	Sec. 5.1.5

Appendix

Table E.1: Evaluation overview.

PDE Operator Learning

Partial differential equation (PDE) operator learning builds models that map inputs such as coefficients and initial conditions to solution fields.

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Neural operators learn mappings between function spaces, enabling fast surrogate solutions to partial differential equations. Despite remarkable architectural diversity, these methods often share a common input representation: raw coordinate-value pairs

(x, u(x))

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