Data, Numbers, and Geometry: Three Tutorials on Numerical Methods, Machine Learning, and Evaluation

Jessica N. Howard; Yidi Qi; Tomás S. R. Silva

cs.LGOct 5, 2026

Data, Numbers, and Geometry: Three Tutorials on Numerical Methods, Machine Learning, and Evaluation

Authors: Jessica N. Howard, Yidi Qi, Tomás S. R. Silva

Organizations: Instituto de Matemática, Estatística e Computação Científica (IMECC), Universidade Estadual de Campinas (UNICAMP), Rua Sérgio Buarque de Holanda, 651, 13083-859 Campinas, São Paulo, Brazil

Abstract

We present three practical tutorials on numerical computation and machine learning for mathematical research, developed for the DANGER: Data, Numbers, and Geometry workshop held at the Banff International Research Station in April 2026. The first develops a numerical approach to exterior calculus from pointwise evaluations of differential forms, using a flux formulation of the exterior derivative. Examples in Euclidean space and on the sphere illustrate geometric identities, topological features, and the effects of approximation and finite precision. The second examines how mathematical structure guides neural network design through examples involving elliptic curves, quivers, and a boundary value problem. It explores how architectural choices affect learning and uses interval arithmetic to bound the residual of a trained network over the full interval of the boundary value problem. The third addresses the evaluation and presentation of machine learning results, covering performance metrics, statistical uncertainty, classification thresholds, receiver operating characteristic curves, and accessible figure design. Throughout, the tutorials distinguish numerical agreement, predictive accuracy, structural guarantees, and rigorous bounds as different forms of evidence. Each contribution can be read independently, with accompanying notebooks and exercises that allow readers to reproduce the examples and adapt the methods to other problems.

Figures & tables

Panel of Figure 1 — Figure 1 : Truncation vs. cancellation for the centered stencil of Eq. ( 3.1 ). The optimum $\varepsilon^{\ast}\approx 10^{-5}$ balances the two regimes; under oracle noise of amplitude $\sigma$ it shifts to $\varepsilon^{\ast}\sim\sigma^{1/3}$ .

	0-form $f$	1-form $\omega$	2-form $\eta$
output of $\mathrm{d}$	1-form $\mathrm{d}f$	2-form $\mathrm{d}\omega$	3-form $\mathrm{d}\eta$
classical name	gradient	curl	divergence
probe region	segment $[a,b]$	parallelogram	parallelepiped
boundary $\partial$	2 oriented points	4 oriented edges	6 oriented faces
boundary integral	$f(b)-f(a)$	$\displaystyle\sum_{\rm edges}\int\omega$	$\displaystyle\sum_{\rm faces}\iint\eta$
$\mathrm{d}^{2}=0$ says…	$\curl(\grad f)=0$	$\dive(\curl\omega)=0$	(top degree)

Table 1 : The unified picture in

\mathbb{R}^{3}

: one algorithm, three classical operators.

$z_{0}$	surface integral	line integral	$\|{\rm diff}\|$
0.0	6.2831853	6.2831853	$\sim 10^{-8}$
0.5	4.7123890	4.7123890	$\sim 10^{-8}$
0.9	1.1938052	1.1938052	$\sim 10^{-8}$

Table 2 : Stokes’ theorem on geodesic caps

\Sigma=\{z\geq z_{0}\}\subset S^{2}

, verified from black-box evaluations only. Exact value:

2\pi(1-z_{0}^{2})

Data type	Key structure	Architecture(s)	Inductive bias
Tabular	unstructured features	MLP, boosted trees	smoothness; axis-aligned splits
Sequential	order, locality	1D CNN, transformer, SSM	shift equivariance; attention; recurrence
Grid / image	2D spatial structure	2D CNN, ViT, U-Net	2D translation equivariance, locality
Set / multiset	no ordering	DeepSets, set transformer	permutation-invariant aggregation
Graph	nodes, edges, topology	GNN (message passing)	permutation equivariance, locality
Geometric	symmetry group ( $E(3)$ , …)	equivariant networks	group equivariance by construction

Table 1 : The landscape: from the structure of the data to the architecture that encodes it.

Figure 1 : (a) Class-averaged Frobenius traces

a_{p}

over the training split. (b) The same after Hasse normalization,

\tilde{a}_{p}=a_{p}/(2\sqrt{p})

. In both, the two class averages are near mirror images and their separation is largest at small

p

. (c) Two individual curves over the first

100

primes: the per-curve signal is indistinguishable from noise, so the pattern is a property of the sequence , recoverable only in aggregate.

Model	Input	Parameters	Train acc.	Test acc.
1D CNN	$1{,}229$ primes	$39{,}714$	$99.91\%$	$\mathbf{99.88\%}$
MLP	$1{,}229$ primes	$39{,}922$	$100.00\%$	$99.13\%$
Transformer	$100$ primes	$11{,}746$	$99.53\%$	$99.51\%$

Table 2 : Rank classification on

3{,}468

held-out curves. The CNN and MLP have matched parameter budgets and see all

1{,}229

primes; the transformer sees only the first

100

(

p\leq 541

), where attention’s

O(T^{2})

cost is affordable, so its accuracy is not directly comparable to the other two.

Figure 2 : Rescaled per-class saliency of the 1D CNN at three training checkpoints. At epoch 1 the map is unstructured noise. By epoch 4 it has become smooth and antisymmetric between the two classes, with oscillatory structure below

p\approx 4000

. By epoch 7 the sensitivity is sharply concentrated in the small primes and decays towards zero for large

p

: the network has localized the discriminative signal in exactly the regime where the class averages of Fig. 1 separate.

Evaluation set	Vertices $n$	Accuracy
Training set	$7,8$ (seen)	$100\%$
Test set	$9$ (unseen)	$99\%$

Table 3 : Quiver mutation-type classification (type

A

versus type

D

), after eleven epochs. The model is trained on

800

quivers with

n=7,8

and evaluated on

100

quivers with

n=9

, a size never seen in training. Independently of training, the node equivariance of Eq. ( 3.4 ) and the invariance of the graph-level logits both hold with maximum error exactly

0.0

Figure 3 : (a) The trained PINN against the exact solution

\sin(\pi x)

. (b) Training loss. (c) The interval enclosure of the residual

-u_{\theta}^{\prime\prime}-f

over

10^{4}

subintervals (band) against the pointwise residual at subinterval centres (line). The enclosure is tight; the residual oscillates within about

3\times 10^{-2}

across the interior and rises steeply at both endpoints.

Figure 1 : Figure showing how a ROC curve can be constructed from a score distribution. Various threshold values are swept from right to left. For each threshold the TPR (fraction of class B scores falling above the threshold) and TNR (fraction of class A scores falling below the threshold) are measured. These (TPR, TNR) points are collected to form the ROC curve. The points on the ROC curve corresponding to the three thresholds are indicated.

Figure 2 : Example of how a plot can be used to efficiently tell a story. We have modified a figure from Ref. [ 12 ] to convey three key points about the result: (1) theoretical predictions agree closely with the empirical measurements, (2) the losses exhibit semi power-law behavior, (3) the two classification tasks follow quantitatively different trends.

Figure 3 : Figure showing a potential fix for a plot with too many lines . The primary goal of the plot is to show that “2D” methods perform worse than “2D ${}_{\Delta{\rm p}_{\rm T}}$ ” methods across all choices of $N$ . However, the plot is trying to do too many pairwise comparisons at once. Since this performance gain is present for all methods (minOT, QminOT 1%, and QminOT 10%), we can simply choose a representative method (in this case QminOT 1%) and say that we see a similar trend in the other methods in the caption. If comparing the performance of the three methods is also important, one can show this in a separate plot. The data and methods for this example were taken from Ref. [ 13 ] .

Panel of Figure 4 — Figure 4 : Figure showing two examples of residual plots . When you want to show agreement between two quantities (in this case binned data and the best fit function) but also want to show the overall shape, it can be helpful to use a residual plot. In the main plot, we show the fitted function and binned data to give an idea of the agreement and shape of the distribution. However, more information about the fit can be gained by looking at the difference or ratio of the two quantities. This can be shown in a residual plot below the main plot. In this case, it helps accentuate differences happening at the tails of the distribution which would be difficult to notice otherwise.

Explore similar work

May 8, 2026cs.LG

A meshfree exterior calculus for generalizable and data-efficient learning of physics from point clouds

We introduce a meshfree exterior calculus (MEEC) for learning structure-preserving descriptions of physics on point clouds, and use it to build MEEC-Net, a data-efficient surrogate that transfers across resolutions, geometries, and physical parameters. MEEC equips an

\varepsilon

-ball graph with virtual node and edge measures via a single sparse Schur complement solve; the resulting complex satisfies discrete conservation exactly, is end-to-end differentiable in the point positions, and exposes a direct geometry-to-physics link without the mesh-generation step required by conventional structure-preserving discretizations. MEEC-Net learns unknown physics as a shared edge-wise flux law in an SO(

d

)-invariant local frame, so the same kernel produces compatible fluxes on any point cloud whose features lie in the training range. We prove a solution-error bound that splits into discretization and kernel-approximation terms which is independent of problem geometry, explaining the observed transfer from very few examples. We show that single-solution training transfers to unseen geometries, boundary conditions, and physical parameters. On five canonical PDE benchmarks MEEC-Net achieves 1-2 orders of magnitude lower out-of-distribution error than baseline neural-operator approaches. On the SimJEB structural-bracket benchmark it achieves competitive error while using substantially fewer training geometries.

Benjamin D. Shaffer, Brooks Kinch, M. Ani Hsieh +1

University of Pennsylvania Philadelphia, PA 19104 · *Mechanical Engineering and Applied Mechanics

Mesh Generation Point Clouds Partial Differential Equations Time Discretization Domain Adaptation

Ask PDF

May 30, 2024cs.LG

A Computational Tropical Geometry Framework for Neural Networks

We propose a computational tropical geometry framework for the symbolic analysis of neural networks with tropical activations. The number of linear regions of a neural network has been actively studied as a measure of the expressivity of a given architecture. To study these, we work in the setting of tropical geometry---a combinatorial and polyhedral variant of algebraic geometry---where there are known connections between tropical rational maps and feedforward neural networks. We expand this connection by developing concrete computational tools for studying the linear regions of neural networks. We present an algorithm, together with a proof of correctness, which computes the linear regions of a neural network as explicit unions of polyhedra. We further relate the computation of the number of linear regions of a tropical expression to the number of monomials that appear in it, and show how tropical expressions can often be pruned to remove redundant monomials. We introduce the Hoffman constant of a neural network's tropical expression, a geometric quantity that controls the distance from any point in the input space to the farthest linear region. We provide the open source Julia library TropicalNN.jl, which is built on top of the OSCAR computer algebra system and implements the algorithms mentioned above to analyze neural networks symbolically using their tropical representations. We present a set of proof-of-concept computational examples to demonstrate how our tropical geometric theory can be applied to reveal insights on the expressivity of a network architecture.

Paul Lezeau, Thomas Walker, Yueqi Cao +2

Department of Mathematics, Imperial College London, UK · London School of Geometry and Number Theory, UK · Department of Electrical and Computer Engineering, Rice University, USA +2

Neural Network Polytopes Representation Space Symbol-

Ask PDF

Jul 7, 2026cs.LG

Level-Crossing Density as a Mesh-Free High-Frequency Auxiliary Loss for Implicit Neural Representations

The Minkowski functionals of a field's excursion sets -- area, boundary measure, and Euler characteristic -- describe its level-set morphology; the Euler characteristic is the cheapest handle on topology. We derive smooth Monte-Carlo estimators for all three of a continuous neural field, evaluated at scattered points via the co-area formula and Gauss-Bonnet, using only autodiff: no grid, no complex, no persistence. The estimator is accurate to 1-3% against exact topology in 2D and 3D, and costs about 3 ms per iteration where a persistent-homology (PH) loss on a cubical grid costs 650-1000 ms -- a 250x gap. We establish four design rules without which these losses silently fail: a dense level ladder (invariants are flat in the parameters away from transitions), a

C^2

backbone (ReLU nets hide curvature in kinks), the full Minkowski vector (Euler characteristic alone is an alternating sum, gamed by debris-hole cancellation; pricing perimeter closes the channel), and sampling-scale coverage. In 2D the vector-valued cap is the only method in a controlled comparison that both repairs topology (3/3 seeds) and preserves fidelity -- uniform smoothing repairs at 11-17x the fidelity cost, and the Euler term alone repairs nothing. In 3D neural-SDF fitting, however, a failure mode we believe general to any sampled soft topology objective appears: gradient descent adversarially hides topological noise below the sampling density, where the estimator is blind -- spurious-feature counts are invariant to 4x more samples, and closing the window needs cubically many points, erasing the cost advantage. A grid-based PH baseline, whose complex is the evaluation resolution, solves the same benchmark (

4/9

exact; median

b_1

error 1 vs. ours above

10^4

). The 250x cost of persistence is, at present, the price of having no null space. We release estimators, receipts, and benchmarks.

Gunner Levi Howe

Independent Researcher

Neural Fields Data Manifold Density Ratio Estimation Monte Carlo High-Frequency Reconstruction

Ask PDF