Cross-MCP Workflows

One of the most powerful features of the Math-Physics-ML MCP system is the ability to chain tools across different servers, creating sophisticated computational pipelines.

How It Works

Each MCP server produces URI references that can be consumed by other servers:

Math MCP          Quantum MCP         Molecular MCP       Neural MCP
─────────         ───────────         ─────────────       ──────────
array://...   →   potential://...  →  trajectory://...  → model://...
               ↘                   ↗
                 └─ shared data ─┘

Workflow Patterns

Pattern 1: Math → Quantum

Use Math MCP to create custom potential functions, then simulate quantum dynamics:

# Step 1: Create custom potential with Math MCP
# Double-well potential for quantum tunneling studies
potential_array = math_mcp.create_array(
    shape=[512],
    fill_type="function",
    function="0.01*(x-128)**2*(x-384)**2 - 500"
)
# Returns: array://abc123...

# Step 2: Use in Quantum MCP simulation
potential = quantum_mcp.create_custom_potential(
    array_uri="array://abc123...",
    grid_size=[512]
)

# Step 3: Create initial state and simulate
psi = quantum_mcp.create_gaussian_wavepacket(
    grid_size=[512],
    position=[128],
    momentum=[0],
    width=15
)

simulation = quantum_mcp.solve_schrodinger(
    potential=potential["potential_id"],
    initial_state=psi,
    time_steps=5000,
    dt=0.05,
    use_gpu=True
)

Pattern 2: Molecular → Math Analysis

Run molecular simulations, then analyze results with advanced math tools:

# Step 1: Run molecular dynamics simulation
system = molecular_mcp.create_particles(
    n_particles=1000,
    box_size=[20, 20, 20],
    temperature=1.0
)

molecular_mcp.add_potential(
    system_id=system["system_id"],
    potential_type="lennard_jones"
)

trajectory = molecular_mcp.run_md(
    system_id=system["system_id"],
    n_steps=100000,
    dt=0.002
)

# Step 2: Get trajectory data
data = molecular_mcp.get_trajectory(
    trajectory_id=trajectory["trajectory_id"]
)

# Step 3: Perform FFT analysis on velocity autocorrelation
# to extract vibrational frequencies
velocities = data["velocities"]
fft_result = math_mcp.fft(
    array=velocities,
    use_gpu=True
)

# Step 4: Find peaks in frequency spectrum
peaks = math_mcp.find_roots(
    function="derivative of spectrum",
    variables=["frequency"],
    initial_guess=[0.1]
)

Pattern 3: Neural → Quantum Surrogate

Train neural networks on quantum simulation data for fast surrogate models:

# Step 1: Generate quantum simulation dataset
training_data = []
for barrier_height in range(5, 50, 5):
    potential = quantum_mcp.create_custom_potential(
        grid_size=[256],
        function=f"{barrier_height}*exp(-(x-128)**2/200)"
    )

    psi = quantum_mcp.create_gaussian_wavepacket(
        grid_size=[256],
        position=[64],
        momentum=[2.0],
        width=10
    )

    sim = quantum_mcp.solve_schrodinger(
        potential=potential["potential_id"],
        initial_state=psi,
        time_steps=1000,
        dt=0.1
    )

    # Calculate transmission coefficient
    result = quantum_mcp.get_simulation_result(sim["simulation_id"])
    training_data.append({
        "barrier_height": barrier_height,
        "transmission": calculate_transmission(result)
    })

# Step 2: Train neural network to predict transmission
model = neural_mcp.define_model(
    architecture="custom",
    num_classes=1  # Regression output
)

# Custom dataset from quantum simulations
experiment = neural_mcp.train_model(
    model_id=model["model_id"],
    dataset=training_data,  # Custom dataset
    epochs=100
)

# Step 3: Use trained model for fast predictions
# 1000x faster than full quantum simulation!

Pattern 4: Multi-Scale Simulation

Combine molecular and quantum simulations for multi-scale physics:

# Quantum region: Electron density calculation
electron_potential = quantum_mcp.create_lattice_potential(
    lattice_type="square",
    grid_size=[64, 64],
    depth=10,
    spacing=8
)

electron_sim = quantum_mcp.solve_schrodinger_2d(
    potential=electron_potential["potential_id"],
    initial_state=initial_wavefunction,
    time_steps=500,
    dt=0.01
)

# Extract effective potential for classical nuclei
electron_density = quantum_mcp.get_simulation_result(
    electron_sim["simulation_id"]
)

# Classical region: Nuclear dynamics with quantum-derived forces
nuclear_system = molecular_mcp.create_particles(
    n_particles=100,
    box_size=[64, 64, 64],
    temperature=300
)

# Add effective potential from quantum calculation
molecular_mcp.add_potential(
    system_id=nuclear_system["system_id"],
    potential_type="custom",
    potential_data=electron_density["effective_potential"]
)

nuclear_trajectory = molecular_mcp.run_nvt(
    system_id=nuclear_system["system_id"],
    n_steps=10000,
    temperature=300
)

Real-World Applications

Drug Discovery Pipeline

┌─────────────┐     ┌─────────────┐     ┌─────────────┐
│   Math MCP  │     │ Molecular   │     │ Neural MCP  │
│             │     │    MCP      │     │             │
│ Generate    │────▶│ Simulate    │────▶│ Predict     │
│ molecular   │     │ binding     │     │ binding     │
│ geometries  │     │ dynamics    │     │ affinity    │
└─────────────┘     └─────────────┘     └─────────────┘

Materials Science

┌─────────────┐     ┌─────────────┐     ┌─────────────┐
│ Quantum MCP │     │   Math MCP  │     │ Molecular   │
│             │     │             │     │    MCP      │
│ Calculate   │────▶│ Fit force   │────▶│ Large-scale │
│ electronic  │     │ field       │     │ simulation  │
│ structure   │     │ parameters  │     │             │
└─────────────┘     └─────────────┘     └─────────────┘

Machine Learning for Physics

┌─────────────┐     ┌─────────────┐     ┌─────────────┐
│ Quantum/    │     │   Math MCP  │     │ Neural MCP  │
│ Molecular   │     │             │     │             │
│ Generate    │────▶│ Feature     │────▶│ Train       │
│ training    │     │ extraction  │     │ surrogate   │
│ data        │     │ & analysis  │     │ model       │
└─────────────┘     └─────────────┘     └─────────────┘

Best Practices

1. Data Format Compatibility

All MCPs use consistent data formats:

Arrays: NumPy-compatible, serialized as JSON lists
URIs: type://uuid format for referencing objects
Complex numbers: {"real": x, "imag": y} dictionaries

2. Error Propagation

Handle errors at each step:

# Check each step succeeded before proceeding
potential = quantum_mcp.create_custom_potential(...)
if "error" in potential:
    raise RuntimeError(f"Potential creation failed: {potential['error']}")

simulation = quantum_mcp.solve_schrodinger(
    potential=potential["potential_id"],
    ...
)
if "error" in simulation:
    raise RuntimeError(f"Simulation failed: {simulation['error']}")

3. Memory Management

For large workflows, clean up intermediate results:

# Large simulation workflow
for config in configurations:
    # Run simulation
    result = run_simulation(config)

    # Extract only needed data
    summary = extract_summary(result)
    summaries.append(summary)

    # Allow garbage collection of large arrays
    del result

# Analyze summaries
final_result = analyze_summaries(summaries)

4. Parallel Execution

Independent operations can run in parallel:

# These can run simultaneously
import concurrent.futures

with concurrent.futures.ThreadPoolExecutor() as executor:
    # Start all simulations
    futures = [
        executor.submit(quantum_mcp.solve_schrodinger, ...)
        for config in configurations
    ]

    # Collect results
    results = [f.result() for f in futures]

Example: Complete Physics Pipeline

Here's a complete example combining all four MCPs:

# Goal: Study quantum effects on molecular diffusion

# 1. MATH MCP: Create analytical potential
potential_expr = math_mcp.symbolic_simplify(
    expression="a*exp(-(x-x0)**2/(2*s**2)) + b*sin(k*x)"
)

potential_array = math_mcp.create_array(
    shape=[256],
    fill_type="function",
    function="5*exp(-(x-128)**2/100) + 2*sin(0.1*x)"
)

# 2. QUANTUM MCP: Solve for quantum ground state
potential = quantum_mcp.create_custom_potential(
    array_uri=potential_array["array_id"],
    grid_size=[256]
)

# Find ground state via imaginary time evolution
ground_state = quantum_mcp.solve_schrodinger(
    potential=potential["potential_id"],
    initial_state=trial_wavefunction,
    time_steps=1000,
    dt=-0.1j  # Imaginary time
)

# 3. MOLECULAR MCP: Classical dynamics with quantum corrections
system = molecular_mcp.create_particles(
    n_particles=500,
    box_size=[50, 50, 50],
    temperature=1.0
)

# Add quantum-corrected potential
molecular_mcp.add_potential(
    system_id=system["system_id"],
    potential_type="custom",
    quantum_correction=ground_state
)

trajectory = molecular_mcp.run_nvt(
    system_id=system["system_id"],
    n_steps=500000,
    temperature=1.0
)

msd = molecular_mcp.compute_msd(trajectory["trajectory_id"])
diffusion = msd["diffusion_coefficient"]

# 4. NEURAL MCP: Train model to predict diffusion from parameters
training_data = collect_diffusion_data()  # Many simulations

model = neural_mcp.define_model(
    architecture="custom",
    num_classes=1
)

neural_mcp.train_model(
    model_id=model["model_id"],
    dataset=training_data,
    epochs=50
)

# Now predict diffusion 1000x faster!
predicted_D = neural_mcp.predict(model["model_id"], new_params)

This workflow demonstrates the power of combining mathematical analysis, quantum mechanics, classical dynamics, and machine learning in a single unified pipeline.

How It Works​

Workflow Patterns​

Pattern 1: Math → Quantum​

Pattern 2: Molecular → Math Analysis​

Pattern 3: Neural → Quantum Surrogate​

Pattern 4: Multi-Scale Simulation​

Real-World Applications​

Drug Discovery Pipeline​

Materials Science​

Machine Learning for Physics​

Best Practices​

1. Data Format Compatibility​

2. Error Propagation​

3. Memory Management​

4. Parallel Execution​

Example: Complete Physics Pipeline​