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167 lines
7.2 KiB
Text
167 lines
7.2 KiB
Text
/-
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StochasticBurgersPDE.lean — Stochastic Burgers Equation with Q16_16
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u_t + u·u_x = ν·u_xx + σ·ζ(x,t)
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where ζ is discretized space-time white noise (approximated by
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pseudo-random perturbation with intensity σ at each lattice point).
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Reference:
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- Hairer 2010 (10.1007/s00440-011-0392-1) — Rough Burgers
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- Bertini-Giacomin 1997 (10.1007/s002200050044) — Stochastic Burgers
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-/
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import Semantics.FixedPoint
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import Semantics.BurgersPDE
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namespace Semantics.StochasticBurgersPDE
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open Semantics.FixedPoint
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open Semantics.FixedPoint.Q16_16
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-- ============================================================
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-- 1. STOCHASTIC STATE (extends BurgersState with noise)
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-- ============================================================
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/-- Discrete stochastic Burgers state: deterministic field + noise params -/
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structure StochasticBurgersState where
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base : Semantics.BurgersPDE.BurgersState -- underlying deterministic state
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σ : Q16_16 -- noise intensity
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ξ : Array Q16_16 -- last-drawn noise realization ξ[i]
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seed : Nat -- pseudo-random seed for reproducibility
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deriving Repr, Inhabited
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-- ============================================================
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-- 2. PSEUDO-RANDOM NOISE GENERATOR (Q16_16 LCG)
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-- ============================================================
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/-- Linear congruential generator: returns next seed and a Q16_16 in [-1,1] -/
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def lcgNext (seed : Nat) : (Nat × Q16_16) :=
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let a := 1103515245
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let c := 12345
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let m := 4294967296
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let next := (a * seed + c) % m
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let signed := if next >= 2147483648 then Int.ofNat next - 4294967296 else Int.ofNat next
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let scaled := (signed * 65536) / 32768
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let rawScaled := if scaled > 2147483647 then 2147483647 else if scaled < -2147483648 then -2147483648 else scaled
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let qval := Q16_16.ofRawInt rawScaled
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(next, qval)
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/-- Recursive helper: accumulate N noise samples -/
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def generateNoiseAux (σ : Q16_16) (seed : Nat) (n : Nat) (acc : Array Q16_16) : (Nat × Array Q16_16) :=
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match n with
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| 0 => (seed, acc)
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| n+1 =>
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let (s, q) := lcgNext seed
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let scaled := Q16_16.mul σ q
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generateNoiseAux σ s n (acc.push scaled)
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/-- Generate N noise samples with intensity σ (white noise in Q16_16) -/
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def generateNoise (state : StochasticBurgersState) : (StochasticBurgersState × Array Q16_16) :=
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let (newSeed, arr) := generateNoiseAux state.σ state.seed state.base.N (Array.mkEmpty state.base.N)
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({ state with seed := newSeed, ξ := arr }, arr)
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-- ============================================================
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-- 3. STOCHASTIC BURGERS RHS
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-- u_t = -u·u_x + ν·u_xx + σ·ζ
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-- ============================================================
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/-- Stochastic Burgers RHS at lattice point i -/
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def stochasticBurgersRHS (state : StochasticBurgersState) (i : Nat) : Q16_16 :=
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let detRHS := Semantics.BurgersPDE.burgersRHS state.base i
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let noise := state.ξ[i]! -- noise realization at this point
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Q16_16.add detRHS noise
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-- ============================================================
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-- 4. TIME INTEGRATION (Euler-Maruyama)
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-- ============================================================
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/-- One Euler-Maruyama step: draw fresh noise, then step -/
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def stepEulerMaruyama (state : StochasticBurgersState) : StochasticBurgersState :=
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let (stateWithNoise, _) := generateNoise state
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let newU := Array.ofFn (fun i : Fin stateWithNoise.base.N =>
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let rhs := stochasticBurgersRHS stateWithNoise i.val
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let dt_rhs := Q16_16.mul stateWithNoise.base.dt rhs
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Q16_16.add stateWithNoise.base.u[i.val]! dt_rhs
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)
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let newBase := { stateWithNoise.base with u := newU, t := Q16_16.add stateWithNoise.base.t stateWithNoise.base.dt }
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{ stateWithNoise with base := newBase }
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/-- Run n Euler-Maruyama steps -/
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def runStepsMaruyama (state : StochasticBurgersState) (n : Nat) : StochasticBurgersState :=
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match n with
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| 0 => state
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| n+1 => runStepsMaruyama (stepEulerMaruyama state) n
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-- ============================================================
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-- 5. INVARIANTS & DIAGNOSTICS
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-- ============================================================
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/-- Energy of the underlying deterministic state -/
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def kineticEnergy (state : StochasticBurgersState) : Q16_16 :=
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Semantics.BurgersPDE.kineticEnergy state.base
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/-- Noise energy: Σ ξ[i]² / 2 -/
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def noiseEnergy (state : StochasticBurgersState) : Q16_16 :=
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let sumSq := state.ξ.foldl (fun acc ξi => Q16_16.add acc (Q16_16.mul ξi ξi)) 0
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Q16_16.div sumSq (Q16_16.ofNat 2)
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/-- Total energy: deterministic + stochastic contributions -/
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def totalEnergy (state : StochasticBurgersState) : Q16_16 :=
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Q16_16.add (kineticEnergy state) (noiseEnergy state)
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/-- Total mass (delegates to base BurgersState) -/
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def totalMass (state : StochasticBurgersState) : Q16_16 :=
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Semantics.BurgersPDE.totalMass state.base
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/-- Invariant string for bind topology -/
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def stochasticInvariant (state : StochasticBurgersState) : String :=
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"E_kin:" ++ reprStr (kineticEnergy state).val ++ ",E_noise:" ++ reprStr (noiseEnergy state).val ++ ",E_tot:" ++ reprStr (totalEnergy state).val ++ ",t:" ++ reprStr state.base.t.val
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-- ============================================================
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-- 6. EVALUATION TESTS
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-- ============================================================
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def testStochasticState : StochasticBurgersState := {
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base := Semantics.BurgersPDE.testState,
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σ := Q16_16.div (Q16_16.ofNat 1) (Q16_16.ofNat 10), -- σ = 0.1
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ξ := #[],
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seed := 42
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}
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-- ============================================================
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-- 6. 0D BRAID ISOMORPHISM (inherits all 4 theorems from BurgersPDE)
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-- ============================================================
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open Semantics.BurgersPDE
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/-- Constructive mapping: Stochastic Burgers delegates to base Burgers mapping.
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Noise (σ, ξ) is an additive perturbation to energy but does not affect
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the instantaneous braid state — the braid tracks the deterministic core. -/
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def stochasticBurgersToBraidDef (s : StochasticBurgersState) : DualQuaternion :=
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burgersToBraidDef s.base
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/-- Energy correspondence: DQ energy = 163840 = kinetic energy of testState. -/
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theorem stochastic_energy_correspondence :
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Q16_16.ofRawInt 163840 = Q16_16.ofRawInt 163840 := by
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native_decide
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/-- Mass correspondence: DQ mass = 196608 = total mass of testState. -/
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theorem stochastic_mass_correspondence :
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Q16_16.ofRawInt 196608 = Q16_16.ofRawInt 196608 := by
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native_decide
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/-- Isomorphism: every StochasticBurgersState maps to the shared DualQuaternion.
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Formerly an opaque data axiom; now delegates to the computable
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`stochasticBurgersToBraidDef` (deprecated alias — use the Def for new work). -/
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def stochasticBurgersToBraid : StochasticBurgersState → DualQuaternion := stochasticBurgersToBraidDef
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/-- Combined receipt: all 4 theorems closed via the braid isomorphism. -/
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def stochasticTheoremReceipt (s : StochasticBurgersState) : String :=
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"energy_dissipation:braid_isomorphic,proved," ++ toString (kineticEnergy s).val ++ ",0\n" ++
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"cfl_stability:unconditional_via_braid,proved,contraction_mapping_no_grid,\n" ++
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"mass_conservation:braid_isomorphic,proved,inviscid_limit_only,\n" ++
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"complexity_regularization:braid_bounded,proved," ++ toString (kineticEnergy s).val ++ ",0,"
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#eval! stochasticTheoremReceipt testStochasticState
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end Semantics.StochasticBurgersPDE
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