Semiconductor Manufacturing Process: Emerging Mathematical Frontiers

1. Computational Lithography and Inverse Problems

1.1 Inverse Lithography Technology (ILT)

The fundamental problem: Given a desired wafer pattern $I_{\text{target}}(x,y)$, find the optimal mask pattern $M(x',y')$.

Core Mathematical Formulation:

$$ \min_{M} \mathcal{L}(M) = \int \left| I(x,y; M) - I_{\text{target}}(x,y) \right|^2 \, dx \, dy + \lambda \mathcal{R}(M) $$

Where:

• $I(x,y; M)$ = Aerial image intensity on wafer
• $I_{\text{target}}(x,y)$ = Desired pattern intensity
• $\mathcal{R}(M)$ = Regularization term (mask manufacturability)
• $\lambda$ = Regularization parameter

Key Challenges:

• Dimensionality: Full-chip optimization involves $N \sim 10^9$ to $10^{12}$ variables
• Non-convexity: The forward model $I(x,y; M)$ is highly nonlinear
• Ill-posedness: Multiple masks can produce similar images

Hopkins Imaging Model:

$$ I(x,y) = \sum_{k} \left| \int \int H_k(f_x, f_y) \cdot \tilde{M}(f_x, f_y) \cdot e^{2\pi i (f_x x + f_y y)} \, df_x \, df_y \right|^2 $$

Where:

• $H_k(f_x, f_y)$ = Transmission cross-coefficient (TCC) eigenfunctions
• $\tilde{M}(f_x, f_y)$ = Fourier transform of mask transmission

1.2 Source-Mask Optimization (SMO)

Bilinear Optimization Problem:

$$ \min_{S, M} \mathcal{L}(S, M) = \| I(S, M) - I_{\text{target}} \|^2 + \alpha \mathcal{R}_S(S) + \beta \mathcal{R}_M(M) $$

Where:

• $S$ = Source intensity distribution (illumination pupil)
• $M$ = Mask transmission function
• $\mathcal{R}_S$, $\mathcal{R}_M$ = Source and mask regularizers

Alternating Minimization Approach:

1. Fix $S^{(k)}$, solve: $M^{(k+1)} = \arg\min_M \mathcal{L}(S^{(k)}, M)$ 2. Fix $M^{(k+1)}$, solve: $S^{(k+1)} = \arg\min_S \mathcal{L}(S, M^{(k+1)})$ 3. Repeat until convergence

1.3 Stochastic Lithography Effects

At EUV wavelengths ($\lambda = 13.5$ nm), photon shot noise becomes critical.

Photon Statistics:

$$ N_{\text{photons}} \sim \text{Poisson}\left( \frac{E \cdot A}{h u} \right) $$

Where:

• $E$ = Exposure dose (mJ/cm²)
• $A$ = Pixel area
• $h

u$ = Photon energy ($\approx 92$ eV for EUV)

Line Edge Roughness (LER) Model:

$$ \text{LER} = \sqrt{\sigma_{\text{shot}}^2 + \sigma_{\text{resist}}^2 + \sigma_{\text{acid}}^2} $$

Stochastic Resist Development (Stochastic PDE):

$$ \frac{\partial h}{\partial t} = -R(M, I, \xi) + \eta(x, y, t) $$

Where:

• $h(x,y,t)$ = Resist height
• $R$ = Development rate (depends on local deprotection $M$, inhibitor $I$)
• $\eta$ = Spatiotemporal noise term
• $\xi$ = Quenched disorder from shot noise

2. Physics-Informed Machine Learning

2.1 Physics-Informed Neural Networks (PINNs)

Standard PINN Loss Function:

$$ \mathcal{L}_{\text{PINN}} = \mathcal{L}_{\text{data}} + \lambda_{\text{PDE}} \mathcal{L}_{\text{PDE}} + \lambda_{\text{BC}} \mathcal{L}_{\text{BC}} $$

Where:

• $\mathcal{L}_{\text{data}} = \frac{1}{N_d} \sum_{i=1}^{N_d} |u_\theta(x_i) - u_i^{\text{obs}}|^2$
• $\mathcal{L}_{\text{PDE}} = \frac{1}{N_r} \sum_{j=1}^{N_r} |\mathcal{N}u_\theta|^2$
• $\mathcal{L}_{\text{BC}} = \frac{1}{N_b} \sum_{k=1}^{N_b} |\mathcal{B}u_\theta - g_k|^2$

Key Mathematical Questions:

• Approximation Theory: What function classes can $u_\theta$ represent under PDE constraints?
• Generalization Bounds: How does enforcing physics improve out-of-distribution performance?

2.2 Neural Operators

Fourier Neural Operator (FNO):

$$ v_{l+1}(x) = \sigma \left( W_l v_l(x) + \mathcal{F}^{-1}\left( R_l \cdot \mathcal{F}(v_l) \right)(x) \right) $$

Where:

• $\mathcal{F}$, $\mathcal{F}^{-1}$ = Fourier and inverse Fourier transforms
• $R_l$ = Learnable spectral weights
• $W_l$ = Local linear transformation
• $\sigma$ = Activation function

DeepONet Architecture:

$$ G_\theta(u)(y) = \sum_{k=1}^{p} b_k(u; \theta_b) \cdot t_k(y; \theta_t) $$

Where:

• $b_k$ = Branch network outputs (encode input function $u$)
• $t_k$ = Trunk network outputs (encode query location $y$)

2.3 Hybrid Physics-ML Architectures

Residual Learning Framework:

$$ u_{\text{full}}(x) = u_{\text{physics}}(x) + u_{\text{NN}}(x; \theta) $$

Where the neural network learns the "correction" to the physics model:

$$ u_{\text{NN}} \approx u_{\text{true}} - u_{\text{physics}} $$

Constraint: Physics Consistency

$$ \| \mathcal{N}[u_{\text{full}}] \|_2 \leq \epsilon $$

3. High-Dimensional Uncertainty Quantification

3.1 Polynomial Chaos Expansions (PCE)

Generalized PCE Representation:

$$ u(\mathbf{x}, \boldsymbol{\xi}) = \sum_{\boldsymbol{\alpha} \in \mathcal{A}} c_{\boldsymbol{\alpha}}(\mathbf{x}) \Psi_{\boldsymbol{\alpha}}(\boldsymbol{\xi}) $$

Where:

• $\boldsymbol{\xi} = (\xi_1, \ldots, \xi_d)$ = Random variables (process variations)
• $\Psi_{\boldsymbol{\alpha}}$ = Multivariate orthogonal polynomials
• $\boldsymbol{\alpha} = (\alpha_1, \ldots, \alpha_d)$ = Multi-index
• $\mathcal{A}$ = Index set (truncated)

Orthogonality Condition:

$$ \mathbb{E}[\Psi_{\boldsymbol{\alpha}} \Psi_{\boldsymbol{\beta}}] = \int \Psi_{\boldsymbol{\alpha}}(\boldsymbol{\xi}) \Psi_{\boldsymbol{\beta}}(\boldsymbol{\xi}) \rho(\boldsymbol{\xi}) \, d\boldsymbol{\xi} = \delta_{\boldsymbol{\alpha}\boldsymbol{\beta}} $$

Curse of Dimensionality:

• Full tensor product: $|\mathcal{A}| = \binom{d + p}{p} \sim \frac{d^p}{p!}$
• Sparse grids: $|\mathcal{A}| \sim \mathcal{O}(d \cdot (\log d)^{d-1})$

3.2 Rare Event Simulation

Importance Sampling:

$$ P(Y > \gamma) = \mathbb{E}_P[\mathbf{1}_{Y > \gamma}] = \mathbb{E}_Q\left[ \mathbf{1}_{Y > \gamma} \cdot \frac{dP}{dQ} \right] $$

Optimal Tilting Measure:

$$ Q^*(\xi) \propto \mathbf{1}_{Y(\xi) > \gamma} \cdot P(\xi) $$

Large Deviation Principle:

$$ \lim_{n \to \infty} \frac{1}{n} \log P(S_n / n \in A) = -\inf_{x \in A} I(x) $$

Where $I(x)$ is the rate function (Legendre transform of cumulant generating function).

3.3 Distributionally Robust Optimization

Wasserstein Ambiguity Set:

$$ \mathcal{P} = \left\{ Q : W_p(Q, \hat{P}_n) \leq \epsilon \right\} $$

DRO Formulation:

$$ \min_{x} \sup_{Q \in \mathcal{P}} \mathbb{E}_Q[f(x, \xi)] $$

Tractable Reformulation (for linear $f$):

$$ \min_{x} \left\{ \frac{1}{n} \sum_{i=1}^{n} f(x, \hat{\xi}_i) + \epsilon \cdot \| abla_\xi f \|_* \right\} $$

4. Multiscale Mathematics

4.1 Scale Hierarchy in Semiconductor Manufacturing

4.2 Homogenization Theory

Two-Scale Expansion:

$$ u^\epsilon(x) = u_0(x, x/\epsilon) + \epsilon u_1(x, x/\epsilon) + \epsilon^2 u_2(x, x/\epsilon) + \ldots $$

Where $y = x/\epsilon$ is the fast variable.

Cell Problem:

$$ - abla_y \cdot \left( A(y) \left( abla_y \chi^j + \mathbf{e}_j \right) \right) = 0 \quad \text{in } Y $$

Effective (Homogenized) Coefficient:

$$ A^*_{ij} = \frac{1}{|Y|} \int_Y A(y) \left( \mathbf{e}_i + abla_y \chi^i \right) \cdot \left( \mathbf{e}_j + abla_y \chi^j \right) \, dy $$

4.3 Phase Field Methods

Allen-Cahn Equation (Interface Evolution):

$$ \frac{\partial \phi}{\partial t} = -M \frac{\delta \mathcal{F}}{\delta \phi} = M \left( \epsilon^2 abla^2 \phi - f'(\phi) \right) $$

Cahn-Hilliard Equation (Conserved Order Parameter):

$$ \frac{\partial c}{\partial t} = abla \cdot \left( M abla \frac{\delta \mathcal{F}}{\delta c} \right) $$

Free Energy Functional:

$$ \mathcal{F}[\phi] = \int \left( \frac{\epsilon^2}{2} | abla \phi|^2 + f(\phi) \right) dV $$

Where $f(\phi) = \frac{1}{4}(\phi^2 - 1)^2$ (double-well potential).

4.4 Kinetic Monte Carlo (KMC)

Master Equation:

$$ \frac{dP(\sigma, t)}{dt} = \sum_{\sigma'} \left[ W(\sigma' \to \sigma) P(\sigma', t) - W(\sigma \to \sigma') P(\sigma, t) \right] $$

Transition Rates (Arrhenius Form):

$$ W_i = u_0 \exp\left( -\frac{E_a^{(i)}}{k_B T} \right) $$

BKL Algorithm:

1. Calculate total rate: $R_{\text{tot}} = \sum_i W_i$ 2. Select event $i$ with probability: $p_i = W_i / R_{\text{tot}}$ 3. Advance time: $\Delta t = -\frac{\ln(r)}{R_{\text{tot}}}$, where $r \sim U(0,1)$

5. Optimization at Unprecedented Scale

5.1 Bayesian Optimization

Gaussian Process Prior:

$$ f(\mathbf{x}) \sim \mathcal{GP}\left( m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}') \right) $$

Posterior Mean and Variance:

$$ \mu_n(\mathbf{x}) = \mathbf{k}_n(\mathbf{x})^T \mathbf{K}_n^{-1} \mathbf{y}_n $$

$$ \sigma_n^2(\mathbf{x}) = k(\mathbf{x}, \mathbf{x}) - \mathbf{k}_n(\mathbf{x})^T \mathbf{K}_n^{-1} \mathbf{k}_n(\mathbf{x}) $$

Expected Improvement (EI):

$$ \text{EI}(\mathbf{x}) = \mathbb{E}\left[ \max(0, f(\mathbf{x}) - f_{\text{best}}) \right] $$

$$ = \sigma_n(\mathbf{x}) \left[ z \Phi(z) + \phi(z) \right], \quad z = \frac{\mu_n(\mathbf{x}) - f_{\text{best}}}{\sigma_n(\mathbf{x})} $$

5.2 High-Dimensional Extensions

Random Embeddings:

$$ f(\mathbf{x}) \approx g(\mathbf{A}\mathbf{x}), \quad \mathbf{A} \in \mathbb{R}^{d_e \times D}, \quad d_e \ll D $$

Additive Structure:

$$ f(\mathbf{x}) = \sum_{j=1}^{J} f_j(\mathbf{x}_{S_j}) $$

Where $S_j \subset \{1, \ldots, D\}$ are (possibly overlapping) subsets.

Trust Region Bayesian Optimization (TuRBO):

• Maintain local GP models within trust regions
• Expand/contract regions based on success/failure
• Multiple trust regions for multimodal landscapes

5.3 Multi-Objective Optimization

Pareto Optimality:

$\mathbf{x}^*$ is Pareto optimal if $ exists \mathbf{x}$ such that:

$$ f_i(\mathbf{x}) \leq f_i(\mathbf{x}^) \; \forall i \quad \text{and} \quad f_j(\mathbf{x}) < f_j(\mathbf{x}^) \; \text{for some } j $$

Expected Hypervolume Improvement (EHVI):

$$ \text{EHVI}(\mathbf{x}) = \mathbb{E}\left[ \text{HV}(\mathcal{P} \cup \{f(\mathbf{x})\}) - \text{HV}(\mathcal{P}) \right] $$

Where $\mathcal{P}$ is the current Pareto front and HV is the hypervolume indicator.

6. Topological and Geometric Methods

6.1 Persistent Homology

Simplicial Complex Filtration:

$$ \emptyset = K_0 \subseteq K_1 \subseteq K_2 \subseteq \cdots \subseteq K_n = K $$

Persistence Pairs:

For each topological feature (connected component, loop, void):

• Birth time: $b_i$ = scale at which feature appears
• Death time: $d_i$ = scale at which feature disappears
• Persistence: $\text{pers}_i = d_i - b_i$

Persistence Diagram:

$$ \text{Dgm}(K) = \{(b_i, d_i)\}_{i=1}^{N} \subset \mathbb{R}^2 $$

Stability Theorem:

$$ d_B(\text{Dgm}(K), \text{Dgm}(K')) \leq \| f - f' \|_\infty $$

Where $d_B$ is the bottleneck distance.

6.2 Optimal Transport

Monge Problem:

$$ \min_{T: T_\# \mu = u} \int c(x, T(x)) \, d\mu(x) $$

Kantorovich (Relaxed) Formulation:

$$ W_p(\mu, u) = \left( \inf_{\gamma \in \Gamma(\mu, u)} \int |x - y|^p \, d\gamma(x, y) \right)^{1/p} $$

Applications in Semiconductor:

• Comparing wafer defect maps
• Loss functions for lithography optimization
• Generative models for realistic defect distributions

6.3 Curvature-Driven Flows

Mean Curvature Flow:

$$ \frac{\partial \Gamma}{\partial t} = \kappa \mathbf{n} $$

Where $\kappa$ is the mean curvature and $\mathbf{n}$ is the unit normal.

Level Set Formulation:

$$ \frac{\partial \phi}{\partial t} + v_n | abla \phi| = 0 $$

With $v_n = \kappa = abla \cdot \left( \frac{ abla \phi}{| abla \phi|} \right)$.

Surface Diffusion (4th Order):

$$ \frac{\partial \Gamma}{\partial t} = -\Delta_s \kappa \cdot \mathbf{n} $$

Where $\Delta_s$ is the surface Laplacian.

7. Control Theory and Real-Time Optimization

7.1 Run-to-Run Control

State-Space Model:

$$ \mathbf{x}_{k+1} = \mathbf{A} \mathbf{x}_k + \mathbf{B} \mathbf{u}_k + \mathbf{w}_k $$

$$ \mathbf{y}_k = \mathbf{C} \mathbf{x}_k + \mathbf{v}_k $$

EWMA (Exponentially Weighted Moving Average) Controller:

$$ \hat{y}_{k+1} = \lambda y_k + (1 - \lambda) \hat{y}_k $$

$$ u_{k+1} = u_k + \frac{T - \hat{y}_{k+1}}{\beta} $$

Where:

• $T$ = Target value
• $\lambda$ = EWMA weight (0 < λ ≤ 1)
• $\beta$ = Process gain

7.2 Model Predictive Control (MPC)

Optimization Problem at Each Step:

$$ \min_{\mathbf{u}_{0:N-1}} \sum_{k=0}^{N-1} \left[ \| \mathbf{x}_k - \mathbf{x}_{\text{ref}} \|_Q^2 + \| \mathbf{u}_k \|_R^2 \right] + \| \mathbf{x}_N \|_P^2 $$

Subject to:

$$ \mathbf{x}_{k+1} = f(\mathbf{x}_k, \mathbf{u}_k) $$

$$ \mathbf{x}_k \in \mathcal{X}, \quad \mathbf{u}_k \in \mathcal{U} $$

Robust MPC (Tube-Based):

$$ \mathbf{x}_k = \bar{\mathbf{x}}_k + \mathbf{e}_k, \quad \mathbf{e}_k \in \mathcal{E} $$

Where $\bar{\mathbf{x}}_k$ is the nominal trajectory and $\mathcal{E}$ is the robust positively invariant set.

7.3 Kalman Filter

Prediction Step:

$$ \hat{\mathbf{x}}_{k|k-1} = \mathbf{A} \hat{\mathbf{x}}_{k-1|k-1} + \mathbf{B} \mathbf{u}_{k-1} $$

$$ \mathbf{P}_{k|k-1} = \mathbf{A} \mathbf{P}_{k-1|k-1} \mathbf{A}^T + \mathbf{Q} $$

Update Step:

$$ \mathbf{K}_k = \mathbf{P}_{k|k-1} \mathbf{C}^T \left( \mathbf{C} \mathbf{P}_{k|k-1} \mathbf{C}^T + \mathbf{R} \right)^{-1} $$

$$ \hat{\mathbf{x}}_{k|k} = \hat{\mathbf{x}}_{k|k-1} + \mathbf{K}_k \left( \mathbf{y}_k - \mathbf{C} \hat{\mathbf{x}}_{k|k-1} \right) $$

$$ \mathbf{P}_{k|k} = \left( \mathbf{I} - \mathbf{K}_k \mathbf{C} \right) \mathbf{P}_{k|k-1} $$

8. Metrology Inverse Problems

8.1 Scatterometry (Optical CD)

Forward Problem (RCWA):

$$ \frac{\partial}{\partial z} \begin{pmatrix} \mathbf{E}_\perp \\ \mathbf{H}_\perp \end{pmatrix} = \mathbf{M}(z) \begin{pmatrix} \mathbf{E}_\perp \\ \mathbf{H}_\perp \end{pmatrix} $$

Inverse Problem:

$$ \min_{\mathbf{p}} \| \mathbf{S}(\mathbf{p}) - \mathbf{S}_{\text{meas}} \|^2 + \lambda \mathcal{R}(\mathbf{p}) $$

Where:

• $\mathbf{p}$ = Geometric parameters (CD, height, sidewall angle)
• $\mathbf{S}$ = Mueller matrix elements
• $\mathcal{R}$ = Regularizer (e.g., Tikhonov, total variation)

8.2 Phase Retrieval

Measurement Model:

$$ I_m = |\mathcal{A}_m x|^2, \quad m = 1, \ldots, M $$

Wirtinger Flow:

$$ x^{(k+1)} = x^{(k)} - \frac{\mu_k}{M} \sum_{m=1}^{M} \left( |a_m^H x^{(k)}|^2 - I_m \right) a_m a_m^H x^{(k)} $$

Uniqueness Conditions:

For $x \in \mathbb{C}^n$, uniqueness (up to global phase) requires $M \geq 4n - 4$ generic measurements.

8.3 Information-Theoretic Limits

Cramér-Rao Lower Bound:

$$ \text{Var}(\hat{\theta}_i) \geq \left[ \mathbf{I}(\boldsymbol{\theta})^{-1} \right]_{ii} $$

Fisher Information Matrix:

$$ [\mathbf{I}(\boldsymbol{\theta})]_{ij} = -\mathbb{E}\left[ \frac{\partial^2 \log p(y | \boldsymbol{\theta})}{\partial \theta_i \partial \theta_j} \right] $$

Optimal Experimental Design:

$$ \max_{\xi} \Phi(\mathbf{I}(\boldsymbol{\theta}; \xi)) $$

Where $\xi$ = experimental design, $\Phi$ = optimality criterion (D-optimal: $\det(\mathbf{I})$, A-optimal: $\text{tr}(\mathbf{I}^{-1})$)

9. Quantum-Classical Boundaries

9.1 Non-Equilibrium Green's Functions (NEGF)

Dyson Equation:

$$ G^R(E) = \left[ (E + i\eta)I - H - \Sigma^R(E) \right]^{-1} $$

Current Calculation:

$$ I = \frac{2e}{h} \int_{-\infty}^{\infty} T(E) \left[ f_L(E) - f_R(E) \right] dE $$

Transmission Function:

$$ T(E) = \text{Tr}\left[ \Gamma_L G^R \Gamma_R G^A \right] $$

Where $\Gamma_{L,R} = i(\Sigma_{L,R}^R - \Sigma_{L,R}^A)$.

9.2 Density Functional Theory (DFT)

Kohn-Sham Equations:

$$ \left[ -\frac{\hbar^2}{2m} abla^2 + V_{\text{eff}}(\mathbf{r}) \right] \psi_i(\mathbf{r}) = \epsilon_i \psi_i(\mathbf{r}) $$

Effective Potential:

$$ V_{\text{eff}}(\mathbf{r}) = V_{\text{ext}}(\mathbf{r}) + V_H(\mathbf{r}) + V_{xc}(\mathbf{r}) $$

Where:

• $V_{\text{ext}}$ = External (ionic) potential
• $V_H = \int \frac{n(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} d\mathbf{r}'$ = Hartree potential
• $V_{xc} = \frac{\delta E_{xc}[n]}{\delta n}$ = Exchange-correlation potential

9.3 Semiclassical Approximations

WKB Approximation:

$$ \psi(x) \approx \frac{C}{\sqrt{p(x)}} \exp\left( \pm \frac{i}{\hbar} \int^x p(x') \, dx' \right) $$

Where $p(x) = \sqrt{2m(E - V(x))}$.

Validity Criterion:

$$ \left| \frac{d\lambda}{dx} \right| \ll 1, \quad \text{where } \lambda = \frac{h}{p} $$

Tunneling Probability (WKB):

$$ T \approx \exp\left( -\frac{2}{\hbar} \int_{x_1}^{x_2} |p(x)| \, dx \right) $$

10. Graph and Combinatorial Methods

10.1 Design Rule Checking (DRC)

Constraint Satisfaction Problem (CSP):

$$ \forall (i,j) \in E: \; d(p_i, p_j) \geq d_{\min}(t_i, t_j) $$

Where:

• $p_i, p_j$ = Polygon features
• $d$ = Distance function (min spacing, enclosure, etc.)
• $t_i, t_j$ = Layer/feature types

SAT/SMT Encoding:

$$ \bigwedge_{r \in \text{Rules}} \bigwedge_{(i,j) \in \text{Violations}(r)} eg(x_i \land x_j) $$

10.2 Graph Neural Networks for Layout

Message Passing Framework:

$$ \mathbf{h}_v^{(k+1)} = \text{UPDATE}^{(k)} \left( \mathbf{h}_v^{(k)}, \text{AGGREGATE}^{(k)} \left( \left\{ \mathbf{h}_u^{(k)} : u \in \mathcal{N}(v) \right\} \right) \right) $$

Graph Attention:

$$ \alpha_{vu} = \frac{\exp\left( \text{LeakyReLU}(\mathbf{a}^T [\mathbf{W}\mathbf{h}_v \| \mathbf{W}\mathbf{h}_u]) \right)}{\sum_{w \in \mathcal{N}(v)} \exp\left( \text{LeakyReLU}(\mathbf{a}^T [\mathbf{W}\mathbf{h}_v \| \mathbf{W}\mathbf{h}_w]) \right)} $$

$$ \mathbf{h}_v' = \sigma\left( \sum_{u \in \mathcal{N}(v)} \alpha_{vu} \mathbf{W} \mathbf{h}_u \right) $$

10.3 Hypergraph Partitioning

Min-Cut Objective:

$$ \min_{\pi: V \to \{1, \ldots, k\}} \sum_{e \in E} w_e \cdot \mathbf{1}[\text{cut}(e, \pi)] $$

Subject to balance constraints:

$$ \left| |\pi^{-1}(i)| - \frac{|V|}{k} \right| \leq \epsilon \frac{|V|}{k} $$

Cross-Cutting Mathematical Themes

Theme 1: Curse of Dimensionality

Tensor Train Decomposition:

$$ \mathcal{T}(i_1, \ldots, i_d) = G_1(i_1) \cdot G_2(i_2) \cdots G_d(i_d) $$

• Storage: $\mathcal{O}(dnr^2)$ vs. $\mathcal{O}(n^d)$
• Where $r$ = TT-rank

Theme 2: Inverse Problems Framework

$$ \mathbf{y} = \mathcal{A}(\mathbf{x}) + \boldsymbol{\eta} $$

Regularized Solution:

$$ \hat{\mathbf{x}} = \arg\min_{\mathbf{x}} \| \mathbf{y} - \mathcal{A}(\mathbf{x}) \|^2 + \lambda \mathcal{R}(\mathbf{x}) $$

Common regularizers:

• Tikhonov: $\mathcal{R}(\mathbf{x}) = \|\mathbf{x}\|_2^2$
• Total Variation: $\mathcal{R}(\mathbf{x}) = \|

abla \mathbf{x}\|_1$

• Sparsity: $\mathcal{R}(\mathbf{x}) = \|\mathbf{x}\|_1$

Theme 3: Certification and Trust

PAC-Bayes Bound:

$$ \mathbb{E}_{h \sim Q}[L(h)] \leq \mathbb{E}_{h \sim Q}[\hat{L}(h)] + \sqrt{\frac{\text{KL}(Q \| P) + \ln(2\sqrt{n}/\delta)}{2n}} $$

Conformal Prediction:

$$ C(x_{\text{new}}) = \{y : s(x_{\text{new}}, y) \leq \hat{q}\} $$

Where $\hat{q}$ = $(1-\alpha)$-quantile of calibration scores.

Key Notation Summary

abla$ | Gradient operator |

abla^2$, $\Delta$ | Laplacian |

u)$ | $p$-Wasserstein distance |