⚡ Live Sandbox: AI Dev Tool: PostgreSQL Vector Search Extension

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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS Boilerplate
# Description: A production-ready FastAPI boilerplate for building micro-SaaS applications with Pydantic v2 validation and PostgreSQL Vector Search Extension.

from fastapi import FastAPI, HTTPException
from fastapi.responses import JSONResponse
from pydantic import BaseModel, validator
from typing import List, Optional
import psycopg2
import psycopg2.extras
import json

# Initialize FastAPI application
app = FastAPI()

# Define Pydantic models for request and response validation
class UserRequest(BaseModel):
    """User request model"""
    id: int
    name: str
    email: str

@validator('email')
    def validate_email(cls, v):
        """Validate email address"""
        if '@' not in v:
            raise ValueError('Invalid email address')
        return v

class UserResponse(BaseModel):
    """User response model"""
    id: int
    name: str
    email: str

# Define PostgreSQL database connection settings
DB_HOST = 'localhost'
DB_NAME = 'mydatabase'
DB_USER = 'myuser'
DB_PASSWORD = 'mypassword'

# Establish database connection
def get_db

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