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Knowledge Amalgamation for Computational Science and Engineering

Theresa Pollinger, Michael Kohlhase, and Harald Köstler Computer Science, FAU Erlangen-Nürnberg August 15, 2018

Outline

Introduction: CSE? Preliminaries A Running Example Theory Graphs Creating ExaSlang Layer 0 MOSIS: Combining MaMoReD and ExaStencils Amalgamating the Model between Theory and Application MOSIS 1.0: Implementation of MOSIS Conclusion

Introduction: CSE?

CSE?

Computational Science and Engineering: Simulation as the “third mode of discovery”

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 1

CSE?

Computational Science and Engineering: Simulation as the “third mode of discovery”

Application Domain Simulations Expertise Numerics Research Simulations Practice Domain Knowledge Model Knowledge Simulations Knowledge

Sub-Disciplines in CSE and Competencies

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 1

CSE?

Computational Science and Engineering: Simulation as the “third mode of discovery”

2D thermal simulation results, source: [HTf]

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 2

The Gap Between Informal PDE Theory and Simulations Practice: It’s always the same questions. . .

Domain expert

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 3

The Gap Between Informal PDE Theory and Simulations Practice: It’s always the same questions. . .

Domain expert Simulations expert

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 3

The Gap Between Informal PDE Theory and Simulations Practice: It’s always the same questions. . .

Domain expert

Did the user enter everything correctly? Have they fully specified the problem they want to solve? Can there even exist a sensible solution to this problem? Can it be obtained with the chosen method? What do they need to do to get their results?

Simulations expert

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 3

Our Answer: Automating the Knowledge Amalgamation based on MaMoReD!

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 4

Our Answer: Automating the Knowledge Amalgamation based on MaMoReD!

Mathematical Models as Research Data:

• the mathematical model,
• all its assumptions,
• and the mathematical background in whose terms it is defined,
• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 4

Our Answer: Automating the Knowledge Amalgamation based on MaMoReD!

Mathematical Models as Research Data:

• the mathematical model,
• all its assumptions,
• and the mathematical background in whose terms it is defined,

are research data in their own right, and as such are represented as as a flexiformal theory graph

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 4

Preliminaries

A Running Example

An engineer who wants to simulate the heat in the walls of her house

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 5

A Running Example

An engineer who wants to simulate the heat in the walls of her house Her friend, a simulations expert

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 5

A Running Example

An engineer who wants to simulate the heat in the walls of her house

a b k1 k2 x

One-dimensional heat conduction problem Her friend, a simulations expert

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 5

The (Static) Heat Equation

(heat equation)

         ρ cp ∂T

∂t −(∇·(k ∇T)) = ˙

qV in Ω T = T0 in Ω at t = 0 T = T ′

• n ∂Ω

with

ρ

the mass density of the material cp the specific heat capacity k the thermal conductivity

˙

qV the volumetric heat flux / “heat sources” in the material T ′ the temperature profile at the boundary T0 the initial temperature distribution

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 6

The (Static) Heat Equation

−∇·(k∇T) = ˙

qV in Ω T = T ′

• n ∂Ω

with k the thermal conductivity

˙

qV the volumetric heat flux / “heat sources” in the material T ′ the temperature profile at the boundary

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 6

The (Static) Heat Equation

−∇·(k∇T) = ˙

qV in Ω T = T ′

• n ∂Ω

with k the thermal conductivity

˙

qV the volumetric heat flux / “heat sources” in the material T ′ the temperature profile at the boundary ...basically gives us a Poisson equation with Dirichlet boundary conditions (Poisson Equation)

• −∆u = f

in Ω u(x) = u′

• n ∂Ω,

. . . which is always uniquely solvable [KA00]

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 6

...and its Solution

f = ˙ qV = sin(x ·π)

   −∆u = sin(x ·π)

in (0,1) u(0) = 1 u(1) = 0 a b k1 k2 x

One-dimensional heat conduction problem

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Heat equation solution example

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 7

...and its Solution

u = sin(x ·π)

π2 − x + 1    −∆u = sin(x ·π)

in (0,1) u(0) = 1 u(1) = 0 a b k1 k2 x

One-dimensional heat conduction problem

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Heat equation solution example

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 7

The ExaStencils Framework for Stencil Codes

Stencil Code: Algorithm that can be expressed as stencils, e. g., most FD schemes ExaStencils: Code that generates highly optimized stencil solvers (itself written in Scala) [Kro+17] ExaSlang: domain specific language (DSL) for the description of the problem to be simulated, and for the solver to be used [KK16]

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 8

The ExaStencils Framework for Stencil Codes

Stencil Code: Algorithm that can be expressed as stencils, e. g., most FD schemes ExaStencils: Code that generates highly optimized stencil solvers (itself written in Scala) [Kro+17] ExaSlang: domain specific language (DSL) for the description of the problem to be simulated, and for the solver to be used [KK16]

Layer 1 : Continuous model Layer 2 : Discretization Layer 3: Solution algorithm Layer 4: Application specification

The ExaSlang language stack

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 8

The ExaStencils Framework for Stencil Codes

Stencil Code: Algorithm that can be expressed as stencils, e. g., most FD schemes ExaStencils: Code that generates highly optimized stencil solvers (itself written in Scala) [Kro+17] ExaSlang: domain specific language (DSL) for the description of the problem to be simulated, and for the solver to be used [KK16]

Layer 1 : Continuous model Layer 2 : Discretization Layer 3: Solution algorithm Layer 4: Application specification

The ExaSlang language stack

But there is no

way of detecting whether the input is invalid (over-/underspecified, conflicting...) interactivity

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 8

Theory Graphs

Theory Morphisms

Inclusion: transports information, “inheritance” View: maps symbols to apply concept of reasoning, “example”, “duck typing”, “refinement”, “implementation” Structure: copies information, allows renaming of symbols, “instantiation” Meta Theory Relation (too meta for us now) [Koh14]

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 9

Theory Graphs

Spatial Domain

Ω : type,n : N Ω_in_Rn : ⊢ Ω ⊂ Rn Ω_sc : ⊢ Ω is connected Rn

. . . Topology . . . Differential Operators

∆ : {m : N}(Ω → Rm) → (Ω → R)

. . . Calculus . . . Unknown unknown_type =

Ω → Rm

Source f : Ω → R Poisson’s Equation PE = λu : unknown_type.∆u .

= f

A theory graph example: The Static Heat Equation as Poisson’s Equation

Theory Graphs

Spatial Domain

Ω : type,n : N Ω_in_Rn : ⊢ Ω ⊂ Rn Ω_sc : ⊢ Ω is connected Rn

. . . Topology . . . Differential Operators

∆ : {m : N}(Ω → Rm) → (Ω → R)

. . . Calculus . . . Unknown unknown_type =

Ω → Rm

Source f : Ω → R Poisson’s Equation PE = λu : unknown_type.∆u .

= f

Cuboid cuboid : Rn → Rn → type # [a;b]×...×[y;z] . . . Wall cross-section over time W : type = [0;1]×[0;1] e :ϕ Differential operators

• n wall cross-section

laplacian_wall : ... f :ψ Temperature t_type = Ω → R Thermal Conductivity k = 2 Volumetric heat flow g = λx.sin(x ·π) Heat equation HE = λT : t_type.( ∂

∂t − k ·∆)T .

= g

Static heat equation static : λT : t_type. ⊢ ∂

∂t T .

= 0

(HE = λT : t_type.− k ·∆T .

= g)

g :χ

ϕ =     

n → 1

Ω → W Ω_in_Rn → ... Ω_sc → ...      χ =     

include f,e unknown_type →t_type f → − g

k

PE → HE

    

A theory graph example: The Static Heat Equation as Poisson’s Equation

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 10

Theory Graphs – Pushouts

S α : type a : α S’ α′ : type a′ : α′ T b : α T’ b′ : α′

φ : α→α′

a→a′

ψ : i→φ

b→b′

i

A very simple pushout setup, source: [OMT]

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 11

MMT

Foundation-independent system for the processing of logics [MMT; RK13], results can be stored and reused using OMDOC

Semigroup formalization in MMT surface syntax, source: [MitM]

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 12

Creating ExaSlang Layer 0 Active Document Theory Graph

Layer 1 : Continuous model Layer 2 : Discretization Layer 3: Solution algorithm Layer 4: Application specification

Layer 0 flexiformal

Schematic of the proposed MOSIS architecture: Models-to-Simulations Interface System

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 13

MOSIS: Combining MaMoReD and ExaStencils

Ephemeral Theories

Concept: Building the theory graph = interview Application knowledge specific to current situation – not reusable

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 14

Ephemeral Theories

Concept: Building the theory graph = interview Application knowledge specific to current situation – not reusable

Ephemeral Theories

Mode of creating theories that are not persistent and gone with every new run of the program (unless we save them) For OMDOC: already in standard for the Symbolic Computation Software Composability Protocol (SCSCP) [Fre+], now an extension to MMT (you can follow up further developments under [MMTJup17])

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 14

Amalgamating the Model between Theory and Application

Connected Subset S : type, S ⊂ V ⊢ V is finite dimensional ⊢ S is connected Topology . . . Connected Subset Boundary ⊢ ∂S is the boundary to S in the topological sense Rn Domain S ⊂ Rn Rn Lebesgue measure: S(⊆ Rn) → R Boundary in Rn ⊢ ∂S = n − 1 dimensional manifold in Rn (Unions of) Cuboids in Rn from, to : Rn predicate S : Rn → bool = [x]∀[i : 1 . . . n]xi ≥ fromi ∧ xi ≤ toi Cuboid Boundary predicate ∂S = [x] . . . Interval from, to : R constructor : R → R → type # [ 1 ; 2 ] n → 1 Interval Boundary predicate ∂S = [x] x . = from ∨ x . = to Codomain codomain : type (usually Rm) Unknown unknowntype = domain → codomain Parameter t : type parameter : t PDE pde : unknowntype → prop

• rder in unknown : PDE

→ unknowntype → N Differential Operators div : . . . . . . Calculus . . . Boundary Conditions Dirichlet : unknowntype → (subset of) ∂S → codomain → prop Neumann : . . . . . . BCs required for PDE measure of BCs required : PDE → ∂S → unknowntype → Lebesgue measure Linear PDE isLinear : ⊢ . . . Elliptic PDE isElliptic : ⊢ . . . Elliptic Linear Dirichlet Boundary Value Problem hasBCsRequired : ⊢ measure of BCs required . = measure of Dirichlet BCs given Functional Analysis . . . Uniquely Solvable PDE uniquelySovable : ⊢ ∃1u . pde(u) . = true Lax Milgram Lemma

. . . . . .

Wall cross-section Ω : type = [0; 1] x : Ω from = 0, to = 1 Inner and outer surface predicate ∂Ω = x.(x . = 0 or x . = 1) Temperature u : Ω → R unknowntype → Ω → R Thermal conductivity α : R = 2.4 Volumetric heat flow f : Ω → R = [x] sin(x · π) Static heat equation −α · ∆u = f(x) Inside and outside temperature u(0) = 20 u(1) = −4 mySolvability ⊢ myPDE is uniquelySolvable Buildings / building components . . . width = 1m Heating . . . Materials . . . material = sandstone First law of thermodynamics . . . Thermostatics . . . Climatology . . . 1 1’ 2b 2a 2a 3 4 4’ Mathematical background knowledge: PDE theory and its Placeholders Ephemeral theories Application domain theory: Engineering and Physics Model domain theory PDE theory Boundary conditions theory Solution theory

Theory graph for PDEs

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 15

Connected Subset S : type, S ⊂ V ⊢ V is finite dimensional ⊢ S is connected Topology . . . Connected Subset Boundary ⊢ ∂S is the boundary to S in the topological sense Rn Domain S ⊂ Rn Rn Lebesgue measure: S(⊆ Rn) → R Boundary in Rn ⊢ ∂S = n − 1 dimensional manifold in Rn (Unions of) Cuboids in Rn from, to : Rn predicate S : Rn → bool = [x]∀[i : 1 . . . n]xi ≥ fromi ∧ xi ≤ toi Cuboid Boundary predicate ∂S = [x] . . . Interval from, to : R constructor : R → R → type # [ 1 ; 2 ] n → 1 Interval Boundary predicate ∂S = [x] x . = from ∨ x . = to Wall cross-section Ω : type = [0; 1] x : Ω from = 0, to = 1 Inner and outer surface predicate ∂Ω = x.(x . = 0 or x . = 1) unkno Thermal α Buildings / building components . . . width = 1m Heating . . . Materials . . . material = sandstone 1 1’ Mathematical background knowledge: PDE theory and its Placeholders Ephemeral theories Application domain theory: Engineering and Physics Model domain theory

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 16

Boundary

• undary to S

sense ⊆ Rn) → R dimensional x] . . . ry [x] . = to Codomain codomain : type (usually Rm) Unknown unknowntype = domain → codomain Parameter t : type parameter : t PDE pde : unknowntype → prop

• rder in unknown : PDE

→ unknowntype → N Differential Operators div : . . . . . . Calculus . . . Bounda Diric (subset co Neumann . . . surface . = 1) Temperature u : Ω → R unknowntype → Ω → R Thermal conductivity α : R = 2.4 Volumetric heat flow f : Ω → R = [x] sin(x · π) Static heat equation −α · ∆u = f(x) Materials . . . material = sandstone First law of thermodynamics . . . Thermostatics . . . Climatology . . . 1’ 2b 2a 2a 3 PDE theory Boundary

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 17

ype → prop wn : PDE → N erators Boundary Conditions Dirichlet : unknowntype → (subset of) ∂S → codomain → prop Neumann : . . . . . . BCs required for PDE measure of BCs required : PDE → ∂S → unknowntype → Lebesgue measure Linear PDE isLinear : ⊢ . . . Elliptic PDE isElliptic : ⊢ . . . Elliptic Linear Dirichlet Boundary Value Problem hasBCsRequired : ⊢ measure of BCs required . = measure of Dirichlet BCs given Functional Analysis . . . Uniquely Solvable PDE uniquelySovable : ⊢ ∃1u . pde(u) . = true Lax Milgram Lemma

. . . . . .

heat equation = f(x) Inside and outside temperature u(0) = 20 u(1) = −4 mySolvability ⊢ myPDE is uniquelySolvable Thermostatics Climatology . . . 3 4 4’ Boundary conditions theory Solution theory

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 18

Realizing MOSIS via Jupyter & MMT

Layer 0 Simulation

Q: What is the domain? A: . . . . . . Q: What are the PDEs? . . . The solution according to ExaStencils looks like this: . . . interview application MMT system Flexiformal and formal background knowledge

Model description

MMT system user configuration files

Layer 1 : Continuous model Layer 2 : Discretization Layer 3: Solution algorithm Layer 4: Application specification

ExaStencils application simulation results q u e r y O K

• m

d

• c

generate

h e l p s d e s i g n produce

Figure: MOSIS 1.0 System Architecture

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 19

MOSIS 1.0: Jupyter Kernel Implementation of MOSIS

Figure: A screenshot of the MOSIS 1.0 Python program, to be found at [PWA]

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 20

MOSIS 1.0 TGView and MPD Integration

Figure: A screenshot of TGView in MOSIS 1.0

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 21

MOSIS 1.0 TGView and MPD Integration

Figure: A screenshot of a TGView MPD in MOSIS 1.0

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 21

Conclusion

Publications MaMoReD: FAIR @ MMS

PDE

Modelica MatLab SBML ExaStencils FEniCS domains 5-dim score

Publications MaMoReD: FAIR @ MMS

PDE

Modelica MatLab SBML ExaStencils FEniCS domains 5-dim score

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 22

Did the user enter everything correctly? Have they fully specified the problem they want to solve? Can there even exist a sensible solution to this problem? Can it be obtained with the chosen method? What do they need to do to get their results?

A knowledge gap

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 23

Application Domain Simulations Expertise Numerics Research Simulations Practice A knowledge gap based on different kinds of knowledge

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 23

Application Domain Simulations Expertise Numerics Research Simulations Practice

MoSIS

A knowledge gap based on different kinds of knowledge MaMoReD:

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 23

Application Domain Simulations Expertise Numerics Research Simulations Practice

MoSIS

A knowledge gap based on different kinds of knowledge MaMoReD: Theory graphs

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 23

Application Domain Simulations Expertise Numerics Research Simulations Practice

MoSIS Active Document Theory Graph

Layer 1 : Continuous model Layer 2 : Discretization Layer 3: Solution algorithm Layer 4: Application specification

Layer 0 flexiformal

A knowledge gap based on different kinds of knowledge MaMoReD: Theory graphs MOSIS as ExaSlang layer 0

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 23

Application Domain Simulations Expertise Numerics Research Simulations Practice

MoSIS Active Document Theory Graph

Layer 1 : Continuous model Layer 2 : Discretization Layer 3: Solution algorithm Layer 4: Application specification

Layer 0 flexiformal

• rder in unknown : PDE

. . . . . .

A knowledge gap based on different kinds of knowledge MaMoReD: Theory graphs MOSIS as ExaSlang layer 0 Model graph grows between theory and application

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 23

Application Domain Simulations Expertise Numerics Research Simulations Practice

MoSIS Active Document Theory Graph

Layer 1 : Continuous model Layer 2 : Discretization Layer 3: Solution algorithm Layer 4: Application specification

Layer 0 flexiformal

• rder in unknown : PDE

. . . . . .

A knowledge gap based on different kinds of knowledge MaMoReD: Theory graphs MOSIS as ExaSlang layer 0 Model graph grows between theory and application MOSIS 1.0: Jupyter kernel & MMT

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 23

We have some building blocks - let’s build tools!

Application Domain Simulations Expertise Numerics Research Simulations Practice

MoSIS Active Document Theory Graph

Layer 1 : Continuous model Layer 2 : Discretization Layer 3: Solution algorithm Layer 4: Application specification

Layer 0 flexiformal

• rder in unknown : PDE

. . . . . .

A knowledge gap based on different kinds of knowledge MaMoReD: Theory graphs MOSIS as ExaSlang layer 0 Model graph grows between theory and application MOSIS 1.0: Jupyter kernel & MMT

• T. Pollinger

| CS, FAU Erlangen-Nürnberg | Knowledge Amalgamation for Computational Science and Engineering August 15, 2018 23