HELIROS

ΨQuantumiℏ∂_tψ = Ĥψ

γUniverseΔs² = c²Δt² − Δx²

∇Deep Learningθ′ = θ − η∇ℒ

π_EImitation Learningℒ_BC = −𝔼 log π_θ

Q*Reinforcement LearningJ = 𝔼∑γ^tr_t

VLAVision · Language · Actiona ∼ π_θ(I,ℓ)

iℏ∂∂t|ψ⟩ = Ĥ|ψ⟩

ψ(x,t) = Ae^i(kx−ωt)

Δx · Δp ≥ ℏ2

E_n = −13.6 eVn²

E = hν = hcλ

p = hλ

γ = 1√(1 − β²)

x′ = γ(x − vt)

t′ = γ(t − vx/c²)

Δs² = c²Δt² − Δx² − Δy² − Δz²

G_μν + Λg_μν = 8πGc⁴T_μν

Attention(Q,K,V) = softmax(QK^T√d_k)V

h^(l) = σ(W^(l)h^(l−1) + b^(l))

ℒ_CE = −∑_i y_i log p_i

θ_t+1 = θ_t − η∇_θℒ

(f * g)[n] = ∑_m f[m]g[n−m]

z = μ + σ ⊙ ε, ε ∼ 𝒩(0,I)

θ* = arg min_θ 𝔼_{(o,a)∼𝒟_E}[−log π_θ(a|o)]

ℒ_BC = −1N∑_i=1^N log π_θ(a_i|o_i)

𝒟 ← 𝒟 ∪ {(s,π_E(s))}

min_π max_D 𝔼_{π_E}[log D] + 𝔼_π[log(1−D)]

D_KL(π_E ∥ π_θ) = ∑ π_E log π_Eπ_θ

J(θ) = 𝔼_{τ∼π_θ}[∑_t=0^H γ^tr_t]

Q*(s,a) = 𝔼[r + γ max_a′Q*(s′,a′)]

∇_θJ = 𝔼[∑_t∇_θlog π_θ(a_t|s_t)Â_t]

δ_t = r_t + γV(s_t+1) − V(s_t)

ℒ^CLIP = 𝔼[min(r_tÂ_t,clip(r_t,1−ε,1+ε)Â_t)]

a_t:t+H ∼ π_θ( · | I_t−k:t, ℓ )

z = [f_v(I_t−k:t); f_ℓ(ℓ)]

p(a_1:H|z) = ∏_h=1^Hp(a_h|a_<h,z)

ℒ_act = ∑_h=1^H ‖a_h − â_h‖²

π_T(a|z) = softmax(logits(z)/T)