Compositional Methods of Engineering Analysis

James Fairbanks

Department of Mechanical & Aerospace Engineering, University of Florida

Fall 2026 Department Seminar

Curriculum Vitae

Mathematics at the University of Florida, then Georgia Tech with NSA, NDSEG, IDA and LLNL support, then the Georgia Tech Research Institute sponsored by DARPA and ONR, then UF's Herbert Wertheim College of Engineering (CISE and MAE) sponsored by DARPA, ONR, ARO, and AFRL

Students, Collaborators, and Family

The presenter with his family in a field of lupines

The Program at a Glance: DIGICAMS for CODAC

DIGICAMS for CODAC research thrusts, outcomes, workforce, and partners

Five interconnected thrusts bridging computational sheaf theory, open systems control, and scalable digital engineering.

A Categorical Architecture for Systems and Control

Categories of Systems, Optimization, Dynamics & Control Systems, and Decisions & Outcomes

Category theory provides the functorial bridge connecting how systems are structured to how they behave and decide.

Sheaves Are Fundamental for Complex Systems

Grids, flow fields, robot teams, and colorings share the same underlying structure:

A/B: Domain Decomposition in FEA mesh and in the smooth solutions

C/D: Interacting subsystems in Multiagent Autonomy and the closed loop dynamics

E/F: A Cellular Sheaf Visualized Geometrically and Algebraically

G: A structured decomposition of a graph and its space of 3-colorings

Seven panels: grid decomposition, PDE field, layered multi-agent domains, vector fields, a cellular sheaf diagram, and graph colorings

PART ONE

Model-Aware Scientific Computing

Syntactic Systems That Compile to Executable Models

WITH

Sophie Libkind

Evan Patterson

Owen Lynch

Kristopher Brown

Benjamin Bumpus

Semantics and Solvers Are Functors

System specification, model semantics, and simulation for an SIR model

Write the model once as structured data. Semantics, solvers, and analyses are functors out of that specification, so a new domain reuses the machinery instead of starting over.

Wiring Diagrams Are Formulas for Systems

Wiring diagrams in circuits, signal processing, chemistry, manufacturing, controls, and engineering design

Every one of these fields already draws boxes and wires. Each drawing is a formula that denotes a system.

Give the notation a formal semantics and the drawing becomes executable.

One Diagram, Two Semantics

1. Optimization Semantics

Boxes are local objective functions; wires share variable coordinates. Composition yields the sum

f(x_1,x_2) + g(x_2,x_3)

2. Dynamical Semantics

Boxes are vector fields; wires couple state coordinates. Composition superposes forces on the shared variable

x_2

Functorial Translation: Gradient flow maps composite optimization directly into composite dynamics.

A wiring diagram read twice: as a composite objective function, and as a composite dynamical system, with gradient descent carrying the first to the second

Composing Optimization Problems

A map phi from two two-element sets onto three variables, and its pullback duplicating the shared coordinate

Share data by copying it. Compose objectives by adding them.

f(x) \;=\; \mathbf{1}^{\top}\Big({\textstyle\bigoplus_i} f_i\Big)(Px) \;=\; {\textstyle\sum_i} f_i(P_i x)

P

copies shared variables to the boxes that use them;

\bigoplus

runs them independently;

\mathbf{1}^{\top}

adds the results.

Contour plots of f on x1 and x2, g on x2 and x3, and their sum on the shared variable

Composing Vector Fields

Share state by copying it. Compose vector fields by adding them.

\dot{x} \;=\; P^{\top}\Big({\textstyle\bigoplus_i} v_i\Big)(Px)

Copy the shared state to each box, run the local fields independently, sum the contributions back.

The source vector fields v and u, each on two variables

The composed field on three variables, shown as slices at fixed z3 and z1

Same diagram, different semantics. The composition rule comes from the diagram, not from the fields.

Euler's Method Is Functorial

Theorem (Functorial Discretization).

Gradient flow

\mathcal{G}\colon \mathbf{Opt} \to \mathbf{DynSys}

and Euler integration

\mathcal{E}_h\colon \mathbf{DynSys} \to \mathbf{DiscreteDynSys}

are monoidal functors:

\mathcal{E}_h(\mathcal{G}(f \oplus g)) = \mathcal{E}_h(\mathcal{G}(f)) \oplus \mathcal{E}_h(\mathcal{G}(g))

Discretizing a composite problem equals composing the discrete solvers. Modular verification is exact.

PART TWO

Computational Physics

Directly Simulating Multiphysics Systems from Hierarchical Specifications

WITH

Luke Morris

Evan Patterson

George Rauta

Matthew Cuffaro

Encoding Equations as Diagrams

●Vertices carry a variable and the space it lives in
●Edges carry the differential operators
●A physical theory becomes a labelled graph, and a graph is something a program can read

Three scalar equations rewritten as a typed diagram of variables and operators — Diffusion, written equation notation vs diagrammatic notation.

The same content, in a form that supports type checking, rewriting, and composition.

Building Multiphysics with Wiring Diagrams

1. Physical Principles (Decapodes)

Fick's law of diffusion, scalar transport by advection, conservation of mass, and superposition of flux, each as a typed diagram

2. Composition Syntax (Wiring Diagram)

An undirected wiring diagram gluing diffusion, advection, flux superposition, and mass conservation along shared variables

Wiring Diagram Semantics

Junctions: Shared variable domains (e.g. ambient space)

Boxes: Physical sub-theories (diffusion, advection)

Wires: Equality constraints on shared ports

Outer Ports: Interface to next level of hierarchy

3. Composed Multiphysics Model

The composed advection-diffusion decapode, with temperature, velocity, and two flux terms

Writing and Composing Decapodes in Julia

Diffusion = @decapode begin
    (C, Ċ)::Form0
    ϕ::Form1

    ϕ  ==  k * d(C)
    Ċ  ==  ⋆(d(⋆(ϕ)))
    ∂ₜ(C) == Ċ
end

DiffusionAdvection = @relation (C, V) begin
    diffusion(C, ϕ₁)
    advection(C, V, ϕ₂)
    superposition(ϕ₁, ϕ₂, ϕ, C)
end

sim = evalsim(apex(oapply(DiffusionAdvection, physics)))
f = sim(mesh, generate)

(C, Ċ)::Form0

declares which differential forms the variables are

d, ⋆

exterior derivative and Hodge star, not finite differences

@relation

the wiring diagram, written as text

evalsim

generates the simulator from the composite

The macro is the syntax; the diagram is what it denotes.

From Advection–Diffusion to Porous Media Convection

Porous_Convection = @decapode begin
  (λ_ρ₀Cp, αρ₀, k_ηf, ϕ)::Constant
  (P, T, Adv, bound_T, bound_Ṫ)::Form0
  (g, qD)::Form1

  bound_T == adiabatic(T)

  # Darcy flux
  ρ == g ∧ (αρ₀ * bound_T)
  P == Δ⁻¹(δ(ρ))
  qD == -k_ηf * (d(P) - ρ)

  Adv == ⋆(interpolate(∧ᵈᵖ₁₁(⋆(d(bound_T)), qD)))
  Ṫ == -1/ϕ * Adv + λ_ρ₀Cp * Δ(bound_T)

  bound_Ṫ == tb_bc(Ṫ)

  ∂ₜ(T) == bound_Ṫ
end

Real multiphysics fits on one screen, and the compiler checks it.

Verified Against the Literature, at Scale

Kelvin–Helmholtz Instability

Rayleigh–Taylor in 3D

Rising Thermal Bubble

Verification Portfolio: Also verified against porous convection, Cahn–Hilliard on GPU, and multigrid convergence on generic subdivision.

Lid-Driven Cavity Flow, Against a Benchmark

Simulated v-velocity across the x-midline and u-velocity across the y-midline at Re = 1000, plotted against the Ghia et al. 1982 benchmark points — Midline velocity profiles at Re = 1000, against Ghia et al. (1982)

Velocity magnitude field with direction arrows, showing the primary vortex and the corner recirculations — Speed and direction of the converged flow

●The generated simulator lands on the benchmark points across both midlines, and recovers the primary vortex with its corner recirculations
●Quantitative agreement, from a specification with no hand-written discretization

PART THREE

Control and Autonomy

Sheaves as the Bridge from Diagrams to Real Robots

WITH

Tyler Hanks

Joana Bou Barcelo

Wilmer Leal

Itay Kadosh

Hans Riess

Matthew Hale

Warren Dixon

One Framework, Three Problems

Optimization

Local objectives glued along shared variables, solved by ADMM or proximal gradient.

Predictive Control

Use reference tracking cost and dynamic constraints over a horizon to do MPC.

Coordination

Minimizing Dirichlet energy of a cellular sheaf drives agents to aligned behavior.

Hierarchy Is Specified, Not Hard-Coded

The fleet is a tree of nested @system blocks: two wings, each built from the same reusable fragments, joined by a relay.

fleet = @nested_system begin
  @system wingA begin
    flight1 = flight(elemsA, :r1)
    charlie = escort(4, 0.6, :t3)
    r2 = relay()
    @link project(flight1, 1) => r2
    @link r2 => centroid(charlie)
  end
  @system wingB begin
    delta = escort(5, 0.7, :t4)
    echo = escort(5, 0.7, :t5)
    r3 = relay()
  end
  r4 = relay()
  @link project(wingA, 1) => r4
  @link r4 => project(wingB, 1)
end

A four-level fleet of nested escort teams tracking orbiting targets, hierarchy tower compiled from the DSL

A four-level fleet — root → wing → flight → escort team — compiled from this DSL. CellularSheaves.jl, AlgebraicJulia.

The spec records the hierarchy; the solver exploits it.

Sheaf Theory Studies Local-to-Global Phenomena

Four sheaves: continuous functions on a space, 3-colorings of a graph, a cellular sheaf of vector spaces, and solutions to a PDE

In every case: data attached locally, agreement conditions where the pieces overlap, and a question about what glues.

Graph-Laplacian Control, and Its Assumptions

Agents are nodes, sensing or communication links are edges. Each agent compares its state with its neighbors', and local disagreement drives the network toward a global objective.

●Compatible state spaces across every agent
●A measurement each pair of neighbors can compare directly
●Two planar robots comparing positions in the same frame

Sixteen agents arranged on a ring communication graph

The agent states on the ring evolving under Laplacian feedback

Natural for consensus, formation control, and leader–follower tracking — as long as the states are comparable.

Real Systems Are Heterogeneous

●Agents and targets need not live in the same dimension
●Only partial or projected measurements may be available
●Sensing models differ across a fleet
●The graph records who talks to whom, and nothing about what they can compare

Drones in the air, surface vessels on the water and submarines below, linked across all three layers — Drones, surface vessels and submarines share a mission but not a state space. Measurements must be translated to be compared.

Cellular Sheaves Over Graphs

Definition (Cellular Sheaf)

A cellular sheaf

\mathcal{F}

over a graph

G = (V, E)

consists of:

A vertex stalk (vector space) $\mathcal{F}(v)$ for each vertex $v \in V$
An edge stalk (vector space) $\mathcal{F}(e)$ for each edge $e \in E$
A linear restriction map $\mathcal{F}_{v \unlhd e}\colon \mathcal{F}(v) \to \mathcal{F}(e)$ for each incident pair $v \unlhd e$

Incidence Matrix vs Exterior Derivative

Graph

Incidence Matrix: One Signed Scalar per Incidence

\delta = \left[\begin{array}{cccccc} & v_1 & v_2 & v_3 & v_4 & v_5 \\ e_1 & -1 & 1 & & & \\ e_2 & & -1 & 1 & & \\ e_3 & 1 & & -1 & & \\ e_4 & -1 & & & 1 & \\ e_5 & & & -1 & 1 & \\ e_6 & & & -1 & & 1 \\ e_7 & & & & -1 & 1 \end{array}\right]

Sheaf

Vector-space stalks drawn as planes standing over the vertices and edges of a base graph

Exterior Derivative: A Block of Restriction Maps per Incidence

\delta = \left[\begin{array}{cccccc} & v_1 & v_2 & v_3 & v_4 & v_5 \\ e_1 & -F_{1 \lhd 12} & F_{2 \lhd 12} & & & \\ e_2 & & -F_{2 \lhd 23} & F_{3 \lhd 23} & & \\ e_3 & -F_{1 \lhd 13} & & F_{3 \lhd 13} & & \\ e_4 & -F_{1 \lhd 14} & & & F_{4 \lhd 14} & \\ e_5 & & & -F_{3 \lhd 34} & F_{4 \lhd 34} & \\ e_6 & & & -F_{3 \lhd 35} & & F_{5 \lhd 35} \\ e_7 & & & & -F_{4 \lhd 45} & F_{5 \lhd 45} \end{array}\right]

Both Laplacians are δ^Tδ; the graph is the sheaf whose restriction maps are all 1.

Three Agents Tracking One Target

Initial configuration: one agent per line, target inside

Equilibrium: each agent at a perpendicular foot

Every agent stalk is ℝ², every edge stalk ℝ: each edge compares the target against the line its agent moves along. This is the sheaf behind a Robotarium demo.

Sheaf Laplacians Compute Global Sections

The sheaf Laplacian is

L_\mathcal{F} \triangleq \delta^\top \delta

where

(\delta \mathbf{x})_{ij} \triangleq \mathcal{F}_{i \unlhd ij} x_i - \mathcal{F}_{j \unlhd ij} x_j

Graph Laplacian:

(L_G \mathbf{x})_i = \sum_{j \in N_i} (x_i - x_j)

Sheaf Laplacian:

(L_\mathcal{F} \mathbf{x})_i = \sum_{j \in N_i} \mathcal{F}_{i \unlhd ij}^\top \left( \mathcal{F}_{i \unlhd ij}(x_i) - \mathcal{F}_{j \unlhd ij}(x_j) \right)

Proposition (Global Sections as Kernel). The space of global sections satisfies

\Gamma(G; \mathcal{F}) \triangleq \ker \delta = \ker L_\mathcal{F}

Theorem (Sheaf Diffusion Convergence). Sheaf diffusion

\dot{\mathbf{x}} = -L_\mathcal{F} \mathbf{x}

converges to the orthogonal projection of

\mathbf{x}(0)

onto

\Gamma(G; \mathcal{F})

[2].

The operator that drives consensus on a graph has an exact analogue for heterogeneous agents, and its kernel is the space of global sections.

Coordination Sheaves

A coordination sheaf provides a precise dictionary between geometry and control objectives.

Definition (Coordination Sheaf). A coordination sheaf on a graph

G=(V,E)

is a cellular sheaf

\mathcal{F}

over

G

together with differentiable potential functions

U_{ij}\colon \mathcal{F}(ij)\to \mathbb{R}

for each edge

ij \in E

MAS Interpretations for Coordination Sheaves

Sheaf Concept	MAS Interpretation
Graph $G$	Bidirectional communication topology
Vertex stalk $\mathcal{F}(i)$	State space of agent $i$
0-cochains $C^0(G;\mathcal{F})$	Global state space
Edge stalk $\mathcal{F}(ij)$	Communication space between $i$ and $j$
Restriction map $\mathcal{F}_{i \unlhd ij}$	Registers state between agents
Potential $U_{ij}$	Coordination goal between $i$ and $j$

Beyond Consensus: Edge Potentials

Minimizing the potential over an edge corresponds to satisfying a coordination goal between the agents incident to that edge.

U(x) = \sum_{e \in E} U_e\big( (\delta x)_e \big)

Let

y \coloneqq \mathcal{F}_{i \unlhd ij} x_i - \mathcal{F}_{j \unlhd ij} x_j

be the edge disagreement between neighbors.

Potential Functions and Coordination Goals

Potential Function	Coordination Goal
$\frac{1}{2}\\|y\\|_2^2$	Consensus
$y^\top A y$	Matrix-weighted consensus
$-\frac{1}{2}\\|y\\|_2^2$	Dissensus
$\frac{1}{2}\\|y - b\\|_2^2$	Reach displacement of $b$
$(\\|y\\|_2^2 - r^2)^2$	Reach distance of $r$

Choosing the edge potential chooses the coordination behavior of the fleet.

Homological Programs and Their Guarantees

Definition (Nonlinear Homological Program). Data

\mathsf{P} = (V, E, \mathcal{F}, \{f_i\}, \{U_e\})

with graph

G

, sheaf

\mathcal{F}

, objectives

\{f_i\}

, and potentials

\{U_e\}

\begin{aligned} \underset{\mathbf{x} \in C^0(G;\mathcal{F})}{\text{minimize}} \quad & \sum_{i \in V} f_i(x_i) \\ \text{subject to} \quad & L_\mathcal{F}^{\nabla U} \mathbf{x} = 0 \end{aligned}

Theorem (Convexity & Tractability). If each edge potential

U_e

is differentiable and convex, and each agent objective

f_i

is convex, then

\mathsf{P}

is a convex optimization problem.

Theorem (Decentralized ADMM Convergence). For convex

\mathsf{P}

with quadratic potentials and a Lagrangian saddle point:

Residual convergence: $\mathbf{x}^k - \mathbf{z}^k \to 0$
Objective convergence: $\sum_i f_i(x_i^k) + \chi_\mathcal{C}(\mathbf{z}^k) \to p^\star$
Dual variable convergence: $\mathbf{y}^k \to \mathbf{y}^\star$ (dual optimal point)

Theorem (Nonlinear Sheaf Diffusion). For single-integrator dynamics

\dot{\mathbf{x}} = \mathbf{u} = -L_\mathcal{F}^{\nabla U}\mathbf{x}

with strongly convex edge potentials, trajectories converge exponentially to

\arg\min_{\mathbf{x} \in C^0(G;\mathcal{F})} U(\delta_\mathcal{F}\mathbf{x})

Local gradient flow on the sheaf potential computes the global coordination state with provable convergence.

Harmonic Extension as Control Objective

Finding a Global Section

\mathcal{L}_{\mathcal{F}} \begin{bmatrix} q \\ p \end{bmatrix} = \begin{bmatrix} \mathcal{H} & -\mathcal{B} \\ -\mathcal{B}^\top & \mathcal{L}_P \end{bmatrix} \begin{bmatrix} q \\ p \end{bmatrix} = 0

Harmonic Extension

\mathcal{H} q = \mathcal{B} p \quad \Longrightarrow \quad q^\star = \mathcal{H}^{-1}\mathcal{B} p

What the Blocks Mean

\mathcal{H} = \mathcal{L}_Q + D

total agent interaction map

\mathcal{B}

agent–target sensing map

\eta = \mathcal{H}q - \mathcal{B}p

agent disagreement relative to targets

Standing Assumption (Control Authority). Target dynamics

\dot{p} = f_p(p, t)

induce motion on the harmonic extension

\dot{q}^\star = \mathcal{H}^{-1}\mathcal{B}\dot{p}

. Agents must have sufficient control authority to keep up with the changing

q^\star(t)

The harmonic extension

q^\star(t)

is the dynamic target reference that agent feedback tracks.

Sheaf-Laplacian Feedback and Its Guarantee

Local Sheaf Disagreement

\eta_i = \sum_{j \in \mathcal{N}_i} \mathcal{F}_{i \unlhd ij}^{\top}\big( \mathcal{F}_{i \unlhd ij} q_i - \mathcal{F}_{j \unlhd ij} x_j \big)

Decentralized Controller

u_i = -k_1\, g_i^{+} \eta_i

Control Gain

k_1 > 0

— a larger gain shrinks the residual tracking error.

Input Map

g_i^{+}

— maps the correction into agent

i

's control space.

Theorem (Semi-global exponential convergence). Under the standing assumptions, the tracking error relative to the harmonic extension converges semi-globally uniformly exponentially to a bounded residual set. A larger gain

k_1

shrinks the residual error arbitrarily.

Residual errors arise solely from dynamics mismatches and bounded disturbances, exactly matching classical singular perturbation bounds.

Each agent evaluates

u_i

solely from neighbour measurements — no global state, no central coordinator.

The ordinary Lyapunov argument for Laplacian feedback, run in the sheaf’s coordinates.

Who Can Track Whom Is a Cohomological Question

Surface vessels on a plane sensing submarines moving in a three-dimensional volume below

Aerial vehicles in a three-dimensional volume sensing surface vessels confined to a plane below

A ship is stuck on the surface, so the target’s three coordinates determine its two. The harmonic extension is unique and the relative cohomology vanishes.

A quadcopter also has an altitude, which nothing it measures about a surface vessel constrains. That spare degree of freedom is the non-vanishing relative cohomology.

Trackability is whether the measurements pin down every degree of freedom the agents have. The cohomology counts the ones they miss.

Stability–Trackability Duality

Stability

Gradient-flow stream plot of a quadratic objective: every trajectory runs into the equilibrium

Harmonic extension

Trackability

No agent sees the whole state; together they leave nothing unconstrained.

●Relative cohomology turns both questions, one analytic and one geometric, into linear algebra on the agent and target subgraphs, answerable by an algorithm rather than by hand
●A control property and a sensing property are the same invariant

Thirteen Agents, Four Targets, Two Domains

●Thirteen planar agents with nonlinear dynamics
●Four underwater targets in a tetrahedron formation on figure-eight paths
●Restriction maps project three-dimensional target states into the plane
●Decentralized control under external disturbances

Each cluster senses one submerged target; dashed links carry agent-to-agent and target-to-target communication.

Surface vehicles track their targets through a projection they can measure; the restriction maps carry the dimension change.

The Agents Reach and Hold the Formation

Formations, in the Plane

Agent and Target Trajectories

Agent orbits above and target figure-eight paths below, in three dimensions

Tracking Error by Gain

Tracking error norm against time for six control gains from k = 0.001 to k = 100

A larger gain shrinks the residual error, exactly matching the convergence bound.

Agents converge from arbitrary initial positions into the square and triangle formations, and hold them as the targets move.

PART FOUR

Toward Compositional Digital Engineering

Carrying the Framework to Hardware and Learning

Layered Control Architecture

1. Harmonic Reference & Velocity Solves (Sheaf)

Computes harmonic position

H q^*(t) = -L_{AB} p(t)

and velocity

H \dot{q}^*(t) = -L_{AB} \dot{p}(t)

via distributed multifrontal tree solve.

2. Joint Tikhonov Filtering (Boundary Layer)

On-board filter

\epsilon \dot{x}_{\text{ref}} = -x_{\text{ref}} + q^*(t)

\epsilon \dot{v}_{\text{ref}} = -v_{\text{ref}} + \dot{q}^*(t)

smooths discrete communication updates.

3. Feedforward + Feedback Control Law

u = -K(x - x_{\text{ref}}) + B^\dagger (v_{\text{ref}} - A_c x)

. Feedforward eliminates dynamic escort tracking lag while preserving input-to-state stability.

Layered escort feedforward simulation showing multi-agent escort formation tracking with feedforward compensation in CellularSheaves.jl — 6-Agent Escort Formation with Feedforward Layered Control in CellularSheaves.jl

Topological coordination at the fleet scale, feedforward tracking at the vehicle scale.

Learning Helps, and Resists Certification

Homogeneous Formation

Heterogeneous Swarm

Learned policies raise performance, and certifying their stability and safety is the hard part. Next: distributed RL techniques inside the sheaf, where the guarantees already live.

From Theorem to Hardware

Next: the same sheaf and the same feedback law on physical differential-drive robots, using the Robotarium at Georgia Tech.

●Eight-agent wheel and star sheaves under wind drift
●Mixed drone and ground-vehicle teams with a three-to-two dimensional projection
●Next: integration with reinforcement learning for unknown agent dynamics

In progress with the Robotarium team and Matt Hale’s lab. Targeting CDC 2027.

Conclusion

Write the system as structured data; the analysis is a functor out of it.

Diagrams Denote Systems

A map from two two-element sets onto three variables, and its pullback duplicating the shared coordinate

Specifications Compile to Solvers

Sheaves Close the Loop

References

[1]B. Fong and D. I. Spivak. An Invitation to Applied Category Theory: Seven Sketches in Compositionality. Cambridge University Press, 2019.
[2]J. Hansen and R. Ghrist. Toward a spectral theory of cellular sheaves. Journal of Applied and Computational Topology, 3(4):315–358, 2019.
[3]J. Hansen and R. Ghrist. Distributed optimization with sheaf homological constraints. Allerton Conference on Communication, Control, and Computing, pp. 565–571, 2019.
[4]T. Hanks, H. Riess, S. Cohen, T. Gross, M. Hale, and J. Fairbanks. Distributed multi-agent coordination over cellular sheaves. arXiv:2504.02049, 2025.
[5]T. Hanks, C. F. Nino, J. Bou Barcelo, A. Copeland, W. E. Dixon, and J. Fairbanks. Heterogeneous multi-agent multi-target tracking using cellular sheaves. European Control Conference. arXiv:2512.24886.

[6]L. Morris, A. Baas, J. Arias, M. Gatlin, E. Patterson, and J. Fairbanks. Decapodes: a diagrammatic tool for representing, composing, and computing spatialized partial differential equations. Journal of Computational Science, 81, 2024.
[7]E. Patterson, O. Lynch, and J. Fairbanks. Categorical data structures for technical computing. Compositionality, 4(5), 2022.
[8]U. Ghia, K. N. Ghia, and C. T. Shin. High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. Journal of Computational Physics, 48(3):387–411, 1982.
[9]S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning, 3(1), 2011.
[10]AlgebraicJulia. CellularSheaves.jl, Decapodes.jl, and CombinatorialSpaces.jl. algebraicjulia.github.io

BACKUP

Additional Material

Slides Held in Reserve for Questions

Beyond Euclidean Gradient Descent

Gradient descent is one functor from optimization problems to dynamical systems. Accelerated methods, saddle-point flows, and Riemannian descent are others, all landing in the same target.

Functors from Euclidean saddle, Euclidean, Riemannian, and general optimization categories into their dynamical counterparts, all landing in a common category of distributed dynamical systems

Tyler Hanks built a distributed algorithm for Riemannian optimization on this structure, extending distributed gradient descent to variables on manifolds.

Why Exact Tracking Fails, and What Replaces It

Exact tracking asks the agent assignment to be a global section. An agent linked to two targets in different states cannot satisfy both edge constraints.

The Canonical Relaxation

Fix targets as boundary values and choose agent states that minimize total disagreement:

x_A^{\star} = \arg\min_{x_A} \; \tfrac12 \big\| \delta \,[\,x_A \, ; \, x_T\,] \big\|^2

Exact consensus is impossible across mismatched domains; the harmonic extension is the canonical, optimal relaxation.

Decapodes: Directly Computable Physics

Discrete exterior calculus supplies the operators; the diagram supplies the equations. The specification compiles to a simulator with no hand-written discretization.

●Specifications are data: type-checked, rewritten, and composed before compilation
●Custom operators supply their own matrices against any mesh
●Generic subdivision gives geometric multigrid for free
●Sparse DEC operators map cleanly onto GPUs

Exterior calculus operators, a discrete geometric domain, physical equations, and the generated simulation

Built on CombinatorialSpaces.jl, part of the AlgebraicJulia ecosystem.

The Spectrum of Scientific Computing Technology

A spectrum from arbitrary code through domain-specific languages and modeling frameworks to computer algebra systems, ordered by how explicitly the model is represented

The compositional program moves right on this axis without giving up generality.

One Structure, Four Activities

Specifications as structured data, composed along interfaces

Modeling

Solvers generated from the specification, verified against the literature

Simulation

Surrogates and calibration constrained by the model's own structure

Learning

Distributed optimization and control read off the same gluing

Decision

Each is usually a separate tool stack with its own model format. Written compositionally they share one representation, so a change to the model propagates to the simulator, the surrogate, and the controller.

That shared representation is what digital engineering needs and what scientific AI keeps discovering it is missing.

Scientific AI Needs the Structure

Learning components constrained by the model they sit inside.

Structured Surrogates

Neural operators and basis expansions parameterized directly over categorical model specifications rather than black-box grids.

Representation

Automated Adjoints

Differentiable calibration and parameter estimation generated automatically by functorial differentiation through the wiring diagram.

Calibration

Model Selection

Searching spaces of morphisms between physical theories to discover missing terms and constitutive relations from observed data.

Discovery

Scientific AI does not replace domain physics—it learns the unknown constitutive terms inside the structure.

Where This Goes Next

Digital engineering built on models that compose.

Autonomous Mission Synthesis

Compiling verified, multi-domain fleet controllers directly from high-level mission specifications and spatio-temporal constraints.

Autonomy

Certified Learning-in-the-Loop

Sheaf-theoretic multi-agent reinforcement learning with provable Lyapunov stability guarantees and Control Barrier Functions on hardware.

Safety & RL

Model-Based Digital Twins

Unifying continuous finite-element multiphysics, reduced-order models, and live hardware sensor streams in one mathematical graph.

Digital Engineering

AlgebraicJulia Ecosystem

Scaling the open-source community into the standard national computational infrastructure for compositional science.

Infrastructure

Compositionality turns complex systems engineering into an algebraic, verifiable discipline.