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 · ScientificComputingLua.md · LifeRL.md · SpaceEngineersRL.md

Goals (all phases)

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

Commands

Differential equations (de)

de status
de catalog list
de catalog describe decay
de catalog run logistic [t1] [dt]
de catalog residual decay
de catalog sde gbm|ou [t1]
de catalog twin focus|burnout|se
de pinn fit decay [steps]
de pinn residual logistic
de dae demo|rc
de dae alg
de linear [t]
de sensitivity [k]
de solve
de rsi | drives | queue | trust | sir
de pde status|demo|heat|wave|poisson|residual
de pde heat [nx] [ny] [k] [t]
de pde wave [nx] [ny] [c] [t]
de pde poisson [nx] [ny] [iters]

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)

algebra demo
algebra quad|roots|system2|eig|binomial|geo|conic|interest|ineq ...
algebra expm [a11 a12 a21 a22]
algebra lsq
algebra lyapunov
algebra gcd | sturm | markov | nullspace | cond

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:

1. Data loss — match catalog RK trajectory samples
2. 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, MathPDE.cpp
• plus existing MathODE, MathCalculus, MathAlgebra, feature front-ends

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

Multi-D PDE (MathPDE)

CPU finite-difference MVP for the multi-D DE integration brainstorm:

ASCII heat-maps print after each run. Residual RMS supports PINN-style collocation checks without full AD.

Deferred

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

Quick smoke checklist

1. de catalog list — shows models including twins + gbm/ou
2. de catalog run decay 2 — final state near e^{-2}
3. de catalog sde gbm 1 — stochastic path
4. de pinn residual decay — finite residual
5. de dae demo — RC discharge
6. calc demo — extended Fourier / Padé / Green lines
7. algebra expm / algebra lyapunov / algebra sturm
8. de pde demo — heat2d + wave2d + poisson2d one-liners
9. de pde heat 32 32 0.2 0.5 — ASCII cooling map