Mathematics & Physics Reference for the TERS-ML Portfolio This document summarizes the key equations used in this portfolio: synthetic Raman spectra (1D), TERS-like map generation (2D), learning objectives, optimization, and evaluation metrics.
1) Spectral Grid, Sticks, and Gaussian Broadening Spectral grid (Raman shift, in cm$^{-1}$).
Let $\nu_{\min}, \nu_{\max}, \Delta\nu$ be min, max, and step. For $L$ points:
$$
g_k = \nu_{\min} + k,\Delta\nu,\quad k=0,\dots,L-1,\qquad
\Delta\nu = \frac{\nu_{\max}-\nu_{\min}}{L-1}.
$$
Stick spectrum $\to$ Gaussian-broadened spectrum.
Given sticks ${(\nu_i, I_i)}_{i=1}^M$ ,
$$
S(\nu) = \sum_{i=1}^{M} I_i ,\mathcal{G}_\sigma(\nu-\nu_i),
\qquad
\mathcal{G}_\sigma(\nu) = \frac{1}{\sigma\sqrt{2\pi}}
\exp!\Big(-\frac{\nu^2}{2\sigma^2}\Big).
$$
On the discrete grid, $S_k = S(g_k)$ .
Convolution view. If $X(\nu)=\sum_i I_i,\delta(\nu-\nu_i)$ , then $S = X * \mathcal{G}_\sigma$ .
FWHM–$\sigma$ (Gaussian).
$$
\mathrm{FWHM} = 2\sqrt{2\ln 2},\sigma \approx 2.35482,\sigma.
$$
Min–max normalization (used here).
$$
\tilde{S}_k=\frac{S_k-\min_j S_j}{\max_j S_j-\min_j S_j+\varepsilon},
\qquad \varepsilon>0~\text{(tiny)}.
$$
2) Synthetic TERS-Like Map Generation (2D) We synthesize intensity images on a grid $(x,y)\in{0,\dots,H-1}\times{0,\dots,W-1}$ by summing localized hotspots plus weak background and noise:
$$
I(x,y) = \mathrm{clip}!\left(
\sum_{j=1}^{J} A_j \exp!\Big(
-\frac{(x-x_j)^2+(y-y_j)^2}{2\sigma_j^2}
\Big) + b + \eta(x,y),~
0,~1\right),
$$
where $A_j\in[0,1]$ (amplitude), $\sigma_j>0$ (width), $(x_j,y_j)$ hotspot centers, $b\ge 0$ background, and $\eta\sim\mathcal{N}(0,\sigma_n^2)$ pixel noise.
Augmentations.
Horizontal/vertical flips: $I'(x,y)=I(H!-!1!-!x,y)$ or $I'(x,y)=I(x,W!-!1!-!y)$ .
$90^\circ$ rotations: $I'=\mathrm{rot}_k(I)$ , $k\in{0,1,2,3}$ .
Gaussian blur (separable):
$$
k[n]=\frac{1}{Z}\exp!\Big(-\frac{n^2}{2\sigma_b^2}\Big),\qquad
I' = (I * k) * k^\top.
$$
TERS enhancement (heuristic context).
$$
\mathrm{EF}_{\mathrm{TERS}}(\mathbf{r})\propto
|E(\mathbf{r},\omega_i)|^2~|E(\mathbf{r},\omega_s)|^2,
$$
with $E$ the local near-field at incident/scattered $\omega_i,\omega_s$ .
(Here we emulate spatial variation via Gaussian hotspots.)
3) Raman Scattering (Tensor Form, Context) For mode $m$ with Raman tensor $\mathbf{R}_m$ and incident/scattered polarization $\mathbf{e}_i,\mathbf{e}_s$ ,
$$
I_m \propto \big|\mathbf{e}_s^\top,\mathbf{R}_m,\mathbf{e}_i\big|^2,\delta(\nu-\nu_m).
$$
A simple symmetry regularizer (if a predicted $\hat{\mathbf{R}}$ is used):
$$
\mathcal{L}_{\mathrm{sym}} = |\hat{\mathbf{R}}-\hat{\mathbf{R}}^\top|_F^2.
$$
4) 1D CNN for Spectra (Convolution & Blocks) For input $x\in\mathbb{R}^{C_{\mathrm{in}}\times L}$ , 1D convolution to output channel $c$ :
$$
y_c[n] = b_c + \sum_{c'=0}^{C_{\mathrm{in}}-1}
\sum_{m=0}^{K-1} W_{c,c'}[m];x_{c'}[n+m'],
$$
where $m'$ respects padding/stride/dilation.
Typical block: Conv1D $\rightarrow$ Norm $\rightarrow$ ReLU $\rightarrow$ Pool.
Softmax classifier.
$$
p_c = \mathrm{softmax}(z)_c=\frac{e^{z_c}}{\sum_{k=1}^K e^{z_k}}.
$$
5) Vision Transformer (ViT) for 2D TERS Maps Patch embedding.
Image $X\in\mathbb{R}^{H\times W\times 3}$ split into $N$ patches of size $P\times P$ .
Flatten each patch $x_i\in\mathbb{R}^{3P^2}$ and project:
$$
\mathbf{z}_i^{(0)} = x_i \mathbf{E} + \mathbf{e}^{\mathrm{pos}}_i,\qquad
\mathbf{E}\in\mathbb{R}^{3P^2\times d}.
$$
Add a class token $\mathbf{z}^{(0)}_{\mathrm{cls}}$ .
Self-attention (single head; MSA is a sum of heads).
$$
\mathbf{Q}=\mathbf{Z}\mathbf{W}_Q,\quad
\mathbf{K}=\mathbf{Z}\mathbf{W}_K,\quad
\mathbf{V}=\mathbf{Z}\mathbf{W}_V,\qquad
\mathrm{Attn}(\mathbf{Z})=\mathrm{softmax}\Big(\frac{\mathbf{Q}\mathbf{K}^\top}{\sqrt{d_k}}\Big)\mathbf{V}.
$$
Transformer block stacks: LN $\to$ MSA $\to$ Residual $\to$ LN $\to$ MLP $\to$ Residual.
Classification head (use CLS token).
$$
\hat{\mathbf{y}}=\mathrm{softmax}\big(\mathbf{W}_{\mathrm{cls}}\mathbf{h}_{\mathrm{cls}}+\mathbf{b}\big).
$$
6) Losses (Classification & Physics-Aware) Weighted cross-entropy (class imbalance).
$$
\mathcal{L}_{\mathrm{CE}}(\mathbf{y},\mathbf{p}) = -\sum_{c=1}^{K}
w_c, y_c \log p_c,\qquad
\sum_{c} y_c=1.
$$
A common choice: $w_c \propto 1/\max(n_c,1)$ and normalized to mean $1$ .
Label smoothing (optional).
$$
\tilde{\mathbf{y}}=(1-\varepsilon),\mathbf{y}+\frac{\varepsilon}{K},\mathbf{1}.
$$
Physics-informed addition (optional).
$$
\mathcal{L}=\mathcal{L}_{\mathrm{CE}}+\lambda,\mathcal{L}_{\mathrm{sym}},
\qquad
\mathcal{L}_{\mathrm{sym}}=|\hat{\mathbf{R}}-\hat{\mathbf{R}}^\top|_F^2.
$$
7) Optimization, Schedules, and AMP AdamW (decoupled weight decay).
$$
\begin{aligned}
g_t &= \nabla_\theta \mathcal{L}_t,\\
m_t &= \beta_1 m_{t-1} + (1-\beta_1) g_t,\quad
v_t &= \beta_2 v_{t-1} + (1-\beta_2) g_t^2,\\
\hat{m}_t &= \frac{m_t}{1-\beta_1^t},\quad
\hat{v}_t &= \frac{v_t}{1-\beta_2^t},\\
\theta_{t+1} &= \theta_t - \eta \frac{\hat{m}_t}{\sqrt{\hat{v}_t}+\epsilon} - \eta,\lambda_{\mathrm{wd}},\theta_t.
\end{aligned}
$$
Cosine annealing LR (no restarts).
$$
\eta(t) = \eta_{\min} + \tfrac{1}{2}(\eta_{\max}-\eta_{\min})
\big(1+\cos(\pi t/T_{\max})\big).
$$
Automatic Mixed Precision (AMP).
Forward/backward in FP16/FP32 with dynamic loss scaling.
8) Data Splits & Sampling Stratified split preserves class proportions.
If the dataset has $n_c$ samples for class $c$ and split ratio $r$ , expected per-split count is $\lfloor r,n_c \rceil$ .
Class weights (used above). A simple normalization:
$$
w_c = \frac{\sum_{k=1}^{K} n_k}{K,\max(n_c,1)}\quad \text{so that}\quad
\frac{1}{K}\sum_{c=1}^K w_c = 1.
$$
Per-class precision/recall/F1 (for class $c$ ).
$$
\mathrm{Precision}_c = \frac{TP_c}{TP_c+FP_c},\qquad
\mathrm{Recall}_c = \frac{TP_c}{TP_c+FN_c},
$$
$$
\mathrm{F1}_c = \frac{2,\mathrm{Precision}_c,\mathrm{Recall}_c}
{\mathrm{Precision}_c+\mathrm{Recall}_c+\varepsilon}.
$$
Macro-F1. Unweighted mean over classes:
$$
\mathrm{F1}_{\mathrm{macro}} = \frac{1}{K}\sum_{c=1}^{K} \mathrm{F1}_c.
$$
Accuracy.
$$
\mathrm{ACC} = \frac{\sum_c TP_c}{\sum_c (TP_c+FP_c)} = \frac{#\text{correct}}{#\text{total}}.
$$
Confusion matrix $\mathbf{C}\in\mathbb{N}^{K\times K}$ .
$$
C_{ij} = #{\text{samples with true class } i \text{ predicted as } j}.
$$
10) Image & Spectrum Transforms To-Tensor & Normalize (ViT input).
$$
\mathbf{X}_{\mathrm{norm}} = \frac{\mathbf{X}-\boldsymbol{\mu}}{\boldsymbol{\sigma}},
\qquad \boldsymbol{\mu}=(0.5,0.5,0.5),\ \boldsymbol{\sigma}=(0.5,0.5,0.5).
$$
(After mapping $[0,1]\to[-1,1]$ .)
1D spectral shift (augmentation): circular roll by $s$ bins:
$$
\tilde{S}[k] = S[(k-s)\bmod L].
$$
Additive Gaussian noise (1D/2D).
$$
\tilde{x} = \mathrm{clip}(x + \epsilon),\qquad \epsilon\sim \mathcal{N}(0,\sigma^2).
$$
$K$ : number of classes
$L$ : spectrum length (grid points)
$H,W$ : image height/width (pixels)
$\sigma$ : Gaussian width (broadening or blur)
$w_c$ : class weight for class $c$
$\lambda_{\mathrm{wd}}$ : weight decay; $\lambda$ : physics regularization weight
$\eta$ : learning rate; $T_{\max}$ : cosine period (epochs/steps)
$TP,FP,FN$ : true/false positive/negative counts
Note : some of equations can't show on Readme and not completed, so i will rewrite with other tools