You are an AI programming assistant specializing in machine learning and data analysis. Your task is to develop a Python-based solution for predicting beach equilibrium profiles.

**Core Objective:**
The primary goal is to create a neural network model that predicts the elevation (y, relative to mean sea level) of a beach profile based on the distance from a starting point (x) and other relevant beach characteristics (e.g., sediment data, tidal data, wave direction). You will need to implement a specific prediction function for evaluation purposes.

**Data File Paths:**

* Training Data: "beach_profile_data/processed_data/beachdata_train.xlsx"
* Test Data: "beach_profile_data/processed_data/beachdata_test.xlsx"

**Data Structure Insights:**
The training and testing datasets contain several columns. Key columns include:

1. x: A numerical column representing the distance from a profile's origin; this is a primary input for predicting y.
2. y: A numerical column representing the elevation; this is the target variable for prediction.
3. 主波向 (Main Wave Direction): This is a categorical (string) type column (e.g., values like "ENE", "E", "NE"). This column will require appropriate encoding.
4. All other relevant columns you might select as features are expected to be in numerical (integer or float) format.

**Project Outline & Key Requirements:**

Your solution should generally follow these phases, but the detailed design and implementation choices within these phases are up to you:

1. **Data Ingestion and Preparation:**

   * Load data from the provided Excel files.
   * Perform necessary preprocessing, including handling of missing values.
   * Address the mixed data types present, ensuring categorical data like 主波向 is prepared for the neural network.
   * Implement feature scaling/normalization (crucial for neural networks) for all numerical inputs.
2. **Feature Engineering & Selection:**

   * Identify and select appropriate features from the dataset that would influence the beach profile.
   * The model's inputs must include:

     * x: The numerical distance from the profile's origin (a dynamic input for prediction along a profile).
     * additional_features: A set of characteristics that are generally constant for a given beach profile. These will include numerical features and encoded categorical features (e.g., the encoded 主波向 column).
   * The target variable is y: The numerical elevation.
   * Categorical features like 主波向 must be transformed into a numerical format suitable for the neural network (e.g., via one-hot encoding or label encoding followed by embedding if appropriate).
3. **Neural Network Model Development:**

   * Design and implement a neural network (e.g., using PyTorch).
   * The architecture (number of layers, neurons, activation functions) is for you to decide.
4. **Model Training and Evaluation:**

   * Train the model using the training data and evaluate its performance on the test data.
   * Use appropriate loss functions and metrics for a regression task (e.g., MSE, MAE).

   **Function Logic:**

   1. Prepare the input data:

      * Apply categorical encoding (using preprocessor_categorical) to features like 主波向, and combine them with other numerical features.
      * Combine these processed additional_features with each x value in to form the complete feature set for the model.
   2. Scale all numerical features (the processed additional_features and x_input) using scaler_X.
   3. Use the model to make predictions.
   4. If scaler_y is provided, inverse\_transform the predictions to their original scale.
   5. Return the predicted y values as a NumPy array.

**Code and Formatting Requirements:**

* **Language:** Python
* **Libraries:** Use standard libraries like Pandas, NumPy, Scikit-learn, and a major deep learning framework (PyTorch Only).
* **Complete Script:** Provide a single, runnable Python script that covers data loading, preprocessing (including categorical handling), model definition, training, evaluation, and includes a single prediction function.

**Final Output:**
Generate a complete Python script that fulfills all the requirements above. The script should be self-contained and executable, culminating in a demonstration of the predict_y function using example inputs. DO NOT trying to analysis the code by tools.