{"id":2008,"date":"2026-07-18T06:41:17","date_gmt":"2026-07-18T13:41:17","guid":{"rendered":"http:\/\/macdaddy4sure.ai\/?p=2008"},"modified":"2026-07-18T06:41:17","modified_gmt":"2026-07-18T13:41:17","slug":"math-de-calculus-algebra-brainstorm-phases","status":"publish","type":"post","link":"http:\/\/macdaddy4sure.ai\/index.php\/2026\/07\/18\/math-de-calculus-algebra-brainstorm-phases\/","title":{"rendered":"Math DE, Calculus & Algebra (Brainstorm Phases)"},"content":{"rendered":"\n

This document describes the differential-equation, calculus, and algebra stack<\/p>\n\n\n\n

added for the DE \/ MathNN \/ DE-kernel brainstorm: named ODE catalog, PINN bridge,<\/p>\n\n\n\n

index-1 DAE lite, domain twins, and extended linear algebra \/ calculus kernels.<\/p>\n\n\n\n

Related: MathNN.md (MathNN.md) \u00b7 ScientificComputingLua.md (ScientificComputingLua.md) \u00b7<\/p>\n\n\n\n

LifeRL.md (LifeRL.md) \u00b7 SpaceEngineersRL.md (SpaceEngineersRL.md)<\/p>\n\n\n\n

Goals (all phases)<\/h2>\n\n\n\n
Phase<\/strong><\/td>Deliverable<\/strong><\/td>Status<\/strong><\/td><\/tr>
1<\/td>ODE\/SDE\/PDE model catalog + MathODE expansions<\/td>Shipped<\/td><\/tr>
2<\/td>Algebra: expm, LSQ, Lyapunov, Sturm, Markov, nullspace<\/td>Shipped<\/td><\/tr>
3<\/td>Calculus: sensitivity, Fourier, Pad\u00e9, directional, mixed, Green<\/td>Shipped<\/td><\/tr>
4<\/td>PINN bridge + DAE lite + domain twins<\/td>Shipped<\/td><\/tr>
5<\/td>Commands, vcxproj, docs<\/td>Shipped<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

True GPU DE kernels, full collocation PINNs with automatic differentiation through<\/p>\n\n\n\n

MathNN, and high-index DAE solvers remain deferred (CPU hybrid MVP first).<\/p>\n\n\n\n

Core files<\/h2>\n\n\n\n
File<\/strong><\/td>Role<\/strong><\/td><\/tr>
MathODE.hpp \/ .cpp<\/td>RK4, RK45, Euler\u2013Maruyama SDE, linear y’=Ay, sensitivity<\/td><\/tr>
MathODECatalog.hpp \/ .cpp<\/td>Named models, twins, residual, SDE demos<\/td><\/tr>
MathPINNBridge.hpp \/ .cpp<\/td>Catalog residual + 1D dense residual+data fit<\/td><\/tr>
MathDAE.hpp \/ .cpp<\/td>Semi-explicit mass-matrix index-1 DAE lite<\/td><\/tr>
MathDEFeatures.hpp \/ .cpp<\/td>User de command surface<\/td><\/tr>
MathCalculus.hpp \/ .cpp<\/td>_CalcKernel (diff, integral, Fourier, Pad\u00e9, Green, \u2026)<\/td><\/tr>
MathCalcFeatures.hpp \/ .cpp<\/td>User calc commands + expression parser<\/td><\/tr>
MathAlgebra.hpp \/ .cpp<\/td>_AlgebraKernel (LA, expm, LSQ, Sturm, Markov, \u2026)<\/td><\/tr>
MathAlgebraFeatures.hpp \/ .cpp<\/td>User algebra commands<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Commands<\/h2>\n\n\n\n

Differential equations (de)<\/h3>\n\n\n\n

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<\/p>\n\n\n\n

Calculus (calc)<\/h3>\n\n\n\n

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]          # \u222b_a^\u221e
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]<\/p>\n\n\n\n

Expressions use x, + – * \/ ^, sin cos tan exp log sqrt abs, pi, e.<\/p>\n\n\n\n

Algebra (algebra)<\/h3>\n\n\n\n

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<\/p>\n\n\n\n

Model catalog<\/h2>\n\n\n\n

Categories: teaching, chaotic, epidemic, control, ai_twin, stochastic, pde_proxy.<\/p>\n\n\n\n

Notable models:<\/p>\n\n\n\n

    \n
  • Teaching:<\/strong> decay, logistic, harmonic, riccati, bernoulli, pendulum, linear2, brusselator<\/li>\n\n\n\n
  • Chaotic:<\/strong> vanderpol, duffing, lorenz, rossler<\/li>\n\n\n\n
  • Epidemic \/ control:<\/strong> sir, lotka<\/li>\n\n\n\n
  • AI twins:<\/strong> focus_twin, burnout_twin, se_power (LifeRL \/ SE power sketches)<\/li>\n\n\n\n
  • Stochastic:<\/strong> gbm, ou (drift ODE + de catalog sde for Euler\u2013Maruyama)<\/li>\n\n\n\n
  • PDE proxy:<\/strong> heat1d (2-cell discrete heat)<\/li>\n<\/ul>\n\n\n\n

    PINN bridge<\/h2>\n\n\n\n

    de pinn fit <model> [steps] trains a small MathNN dense net u_\u03b8(t) with:<\/p>\n\n\n\n

      \n
    1. Data loss<\/strong> \u2014 match catalog RK trajectory samples\u00a0<\/li>\n\n\n\n
    2. Residual loss<\/strong> \u2014 finite-difference u_t vs catalog f(t,u)<\/li>\n<\/ol>\n\n\n\n

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

      bridge, not a full multi-D collocation PINN with automatic differentiation of<\/p>\n\n\n\n

      the ODE residual through all layers.<\/p>\n\n\n\n

      de pinn residual <model> reports catalog trajectory residual RMS (collocation<\/p>\n\n\n\n

      check without NN).<\/p>\n\n\n\n

      DAE lite<\/h2>\n\n\n\n

      Semi-explicit form: diagonal mass matrix M. Zero mass entries are treated as<\/p>\n\n\n\n

      algebraic constraints and projected with a short damped Newton-like update.<\/p>\n\n\n\n

        \n
      • de dae demo \u2014 RC discharge ODE as DAE path\u00a0<\/li>\n\n\n\n
      • de dae alg \u2014 algebraic constraint demo y0’=y1, 0=y0+y1-1<\/li>\n<\/ul>\n\n\n\n

        MathODE expansions<\/h2>\n\n\n\n
          \n
        • LinearStep \/ LinearIntegrate \u2014 dense y’ = A y\u00a0<\/li>\n\n\n\n
        • IntegrateSensitivity \u2014 simultaneous state + dy\/dp via FD Jacobian\u00a0<\/li>\n\n\n\n
        • Existing: RK4, adaptive RK45, Euler\u2013Maruyama, Newton helpers\u00a0<\/li>\n<\/ul>\n\n\n\n

          Algebra \/ calculus kernels<\/h2>\n\n\n\n

          Algebra:<\/strong> LeastSquares, ConditionEstimate, MatrixExp, SolveLyapunov,<\/p>\n\n\n\n

          PolyGcd, Sturm root count, MarkovStationary, NullspaceBasis, MatMul \/<\/p>\n\n\n\n

          Transpose \/ Identity.<\/p>\n\n\n\n

          Calculus:<\/strong> MixedPartial, ImproperIntegralTail, Fourier cos\/sin coeffs,<\/p>\n\n\n\n

          PadeApproximant, LineIntegral, GreenTheoremCheck2D, plus existing gradient \/<\/p>\n\n\n\n

          Hessian \/ Laplacian \/ directional derivative.<\/p>\n\n\n\n

          Lua<\/h2>\n\n\n\n

          Existing bindings remain:<\/p>\n\n\n\n

            \n
          • ai_de_status, ai_de_rsi_observe<\/li>\n\n\n\n
          • ai_calc_eval, ai_calc_integrate, ai_calc_diff<\/li>\n\n\n\n
          • ai_algebra_quadratic, ai_algebra_roots<\/li>\n<\/ul>\n\n\n\n

            Build<\/h2>\n\n\n\n

            Units are in _AugmentedIntelligence.vcxproj:<\/p>\n\n\n\n

              \n
            • MathODECatalog.cpp, MathPINNBridge.cpp, MathDAE.cpp<\/li>\n\n\n\n
            • plus existing MathODE, MathCalculus, MathAlgebra, feature front-ends<\/li>\n<\/ul>\n\n\n\n

              No extra preprocessor flags are required for the DE\/calc\/algebra MVP.<\/p>\n\n\n\n

              Deferred<\/h2>\n\n\n\n
                \n
              • Full CUDA\/OpenCL DE integrators and spectral PDE solvers\u00a0<\/li>\n\n\n\n
              • True automatic-diff PINNs multi-D + boundary loss\u00a0<\/li>\n\n\n\n
              • High-index DAE (Pantelides \/ dummy derivatives)\u00a0<\/li>\n\n\n\n
              • Symbolic algebra \/ CAS\u00a0<\/li>\n\n\n\n
              • Production control LQR from SolveLyapunov \/ Riccati plant catalog\u00a0<\/li>\n<\/ul>\n\n\n\n

                Quick smoke checklist<\/h2>\n\n\n\n
                  \n
                • de catalog list \u2014 shows models including twins + gbm\/ou\u00a0<\/li>\n\n\n\n
                • de catalog run decay 2 \u2014 final state near e^{-2}\u00a0<\/li>\n\n\n\n
                • de catalog sde gbm 1 \u2014 stochastic path\u00a0<\/li>\n\n\n\n
                • de pinn residual decay \u2014 finite residual\u00a0<\/li>\n\n\n\n
                • de dae demo \u2014 RC discharge\u00a0<\/li>\n\n\n\n
                • calc demo \u2014 extended Fourier \/ Pad\u00e9 \/ Green lines\u00a0<\/li>\n\n\n\n
                • algebra expm \/ algebra lyapunov \/ algebra sturm\u00a0<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"

                  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) \u00b7 ScientificComputingLua.md (ScientificComputingLua.md) \u00b7 LifeRL.md (LifeRL.md) \u00b7 SpaceEngineersRL.md (SpaceEngineersRL.md) Goals (all phases) Phase Deliverable Status 1 ODE\/SDE\/PDE […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-2008","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"http:\/\/macdaddy4sure.ai\/index.php\/wp-json\/wp\/v2\/posts\/2008","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/macdaddy4sure.ai\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/macdaddy4sure.ai\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/macdaddy4sure.ai\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/macdaddy4sure.ai\/index.php\/wp-json\/wp\/v2\/comments?post=2008"}],"version-history":[{"count":1,"href":"http:\/\/macdaddy4sure.ai\/index.php\/wp-json\/wp\/v2\/posts\/2008\/revisions"}],"predecessor-version":[{"id":2009,"href":"http:\/\/macdaddy4sure.ai\/index.php\/wp-json\/wp\/v2\/posts\/2008\/revisions\/2009"}],"wp:attachment":[{"href":"http:\/\/macdaddy4sure.ai\/index.php\/wp-json\/wp\/v2\/media?parent=2008"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/macdaddy4sure.ai\/index.php\/wp-json\/wp\/v2\/categories?post=2008"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/macdaddy4sure.ai\/index.php\/wp-json\/wp\/v2\/tags?post=2008"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}