Math DE, Calculus & Algebra (Brainstorm Phases)

This document describes the differential-equation, calculus, and algebra stack

added for the DE / MathNN / DE-kernel brainstorm: named ODE catalog, PINN bridge,

index-1 DAE lite, domain twins, and extended linear algebra / calculus kernels.

Related: MathNN.md (MathNN.md) · ScientificComputingLua.md (ScientificComputingLua.md) ·

LifeRL.md (LifeRL.md) · SpaceEngineersRL.md (SpaceEngineersRL.md)

Goals (all phases)

Phase	Deliverable	Status
1	ODE/SDE/PDE model catalog + MathODE expansions	Shipped
2	Algebra: expm, LSQ, Lyapunov, Sturm, Markov, nullspace	Shipped
3	Calculus: sensitivity, Fourier, Padé, directional, mixed, Green	Shipped
4	PINN bridge + DAE lite + domain twins	Shipped
5	Commands, vcxproj, docs	Shipped

True GPU DE kernels, full collocation PINNs with automatic differentiation through

MathNN, and high-index DAE solvers remain deferred (CPU hybrid MVP first).

Core files

File	Role
MathODE.hpp / .cpp	RK4, RK45, Euler–Maruyama SDE, linear y’=Ay, sensitivity
MathODECatalog.hpp / .cpp	Named models, twins, residual, SDE demos
MathPINNBridge.hpp / .cpp	Catalog residual + 1D dense residual+data fit
MathDAE.hpp / .cpp	Semi-explicit mass-matrix index-1 DAE lite
MathDEFeatures.hpp / .cpp	User de command surface
MathCalculus.hpp / .cpp	_CalcKernel (diff, integral, Fourier, Padé, Green, …)
MathCalcFeatures.hpp / .cpp	User calc commands + expression parser
MathAlgebra.hpp / .cpp	_AlgebraKernel (LA, expm, LSQ, Sturm, Markov, …)
MathAlgebraFeatures.hpp / .cpp	User algebra commands

Commands

Differential equations (de)

Calculus (calc)

calc demo
calc diff <expr> <x>
calc integrate <expr> <a> <b>
calc root|min|optimize|taylor|gradient …
calc fourier <expr> [period] [n_terms]
calc fourier_sin <expr> …
calc pade <expr> [x0] [m] [k]
calc improper <expr> [a] # ∫_a^∞
calc directional [x y ux uy]
calc mixed [x y]
calc laplacian|hessian [x y]
calc green [x0 y0 x1 y1]
calc line [ax ay bx by]
calc sensitivity [a]

Expressions use x, + – * / ^, sin cos tan exp log sqrt abs, pi, e.

Algebra (algebra)

Model catalog

Categories: teaching, chaotic, epidemic, control, ai_twin, stochastic, pde_proxy.

Notable models:

Teaching: decay, logistic, harmonic, riccati, bernoulli, pendulum, linear2, brusselator
Chaotic: vanderpol, duffing, lorenz, rossler
Epidemic / control: sir, lotka
AI twins: focus_twin, burnout_twin, se_power (LifeRL / SE power sketches)
Stochastic: gbm, ou (drift ODE + de catalog sde for Euler–Maruyama)
PDE proxy: heat1d (2-cell discrete heat)

PINN bridge

de pinn fit <model> [steps] trains a small MathNN dense net u_θ(t) with:

Data loss — match catalog RK trajectory samples
Residual loss — finite-difference u_t vs catalog f(t,u)

Checkpoints go under checkpoints/pinn_<model>.net. This is a lite residual

bridge, not a full multi-D collocation PINN with automatic differentiation of

the ODE residual through all layers.

de pinn residual <model> reports catalog trajectory residual RMS (collocation

check without NN).

DAE lite

Semi-explicit form: diagonal mass matrix M. Zero mass entries are treated as

algebraic constraints and projected with a short damped Newton-like update.

de dae demo — RC discharge ODE as DAE path
de dae alg — algebraic constraint demo y0’=y1, 0=y0+y1-1

MathODE expansions

LinearStep / LinearIntegrate — dense y’ = A y
IntegrateSensitivity — simultaneous state + dy/dp via FD Jacobian
Existing: RK4, adaptive RK45, Euler–Maruyama, Newton helpers

Algebra / calculus kernels

Algebra: LeastSquares, ConditionEstimate, MatrixExp, SolveLyapunov,

PolyGcd, Sturm root count, MarkovStationary, NullspaceBasis, MatMul /

Transpose / Identity.

Calculus: MixedPartial, ImproperIntegralTail, Fourier cos/sin coeffs,

PadeApproximant, LineIntegral, GreenTheoremCheck2D, plus existing gradient /

Hessian / Laplacian / directional derivative.

Lua

Existing bindings remain:

ai_de_status, ai_de_rsi_observe
ai_calc_eval, ai_calc_integrate, ai_calc_diff
ai_algebra_quadratic, ai_algebra_roots

Build

Units are in _AugmentedIntelligence.vcxproj:

MathODECatalog.cpp, MathPINNBridge.cpp, MathDAE.cpp
plus existing MathODE, MathCalculus, MathAlgebra, feature front-ends

No extra preprocessor flags are required for the DE/calc/algebra MVP.

Deferred

Full CUDA/OpenCL DE integrators and spectral PDE solvers
True automatic-diff PINNs multi-D + boundary loss
High-index DAE (Pantelides / dummy derivatives)
Symbolic algebra / CAS
Production control LQR from SolveLyapunov / Riccati plant catalog

Quick smoke checklist

de catalog list — shows models including twins + gbm/ou
de catalog run decay 2 — final state near e^{-2}
de catalog sde gbm 1 — stochastic path
de pinn residual decay — finite residual
de dae demo — RC discharge
calc demo — extended Fourier / Padé / Green lines
algebra expm / algebra lyapunov / algebra sturm

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