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Engineering Examples

The examples/engineering/ directory contains real-world engineering applications of stochastic differential equations, organized in tiers of increasing complexity.

Running Engineering Examples

# List engineering examples
uv run asgard example --list | grep engineering

# Run all engineering examples
uv run asgard example -c engineering

# Run a single example
uv run asgard example engineering/01_rc_circuit_thermal_noise.yaml

# Run with monitor dashboard
uv run asgard example -c engineering --monitor

Tier Organization

Tier 1 — Ornstein-Uhlenbeck (Constant Diffusion)

Simple mean-reverting processes with constant noise intensity. These are the building blocks for more complex models.

# Example Domain SDE
01 RC Circuit Thermal Noise Electrical dV = -(V/RC)dt + sigma dW
02 Newton's Cooling Thermal dT = -k(T - T_env)dt + sigma dW
03 First-Order Control Control dx = -a(x - x_ref)dt + sigma dW

Common structure: dX = -alpha(X - mu)dt + sigma dW — linear mean reversion to equilibrium with additive noise.

Tier 2 — Geometric Brownian Motion (Multiplicative Noise)

Processes where noise scales with the state value — common in growth and decay models.

# Example Domain SDE
04 Population Dynamics Ecology dN = rN dt + sigmaN dW
05 Component Degradation Mechanical dD = muD dt + sigmaD dW
06 Stochastic Inflation Financial dP = muP dt + sigmaP dW

Common structure: dX = mu*X dt + sigma*X dW — exponential growth/decay with proportional noise. Solutions are always positive (lognormal distribution).

Tier 3 — Square-Root and Absolute-Value Diffusion

Processes with state-dependent noise that requires special handling.

# Example Domain Feature
07 Chemical Birth-Death Chemical sqrt() diffusion (CLE)
08 Structural Vibration Structural abs() in diffusion

Key insight: The chemical Langevin equation uses sqrt(X) diffusion, which naturally arises from the Poisson nature of molecular reactions. The abs() guard prevents NaN from transiently negative states.

Tier 4 — Coupled SDEs

Multi-equation systems where state variables interact through their drift terms.

# Example Domain Coupling
09 Mass-Spring-Damper Mechanical Position + velocity
10 Predator-Prey (Lotka-Volterra) Ecology Bilinear interaction
11 Kalman-Bucy State Estimation Control Hidden state + observation

See the Coupled SDEs guide for the YAML format and expression syntax.

Tier 5 — Decision and Optimization

These examples go beyond simulation to answer actionable questions.

# Example Domain Key Feature
12 Epidemic Outbreak Response (SIR) Public Health 3-coupled SDE, time-dependent intervention
13 Infrastructure Budget Contingency Finance Percentile analysis for contingency sizing
14 Predictive Maintenance Scheduling Reliability First-passage time for maintenance scheduling
15 Spring Constant Identification Mechanical Gradient optimization, inverse problem

What makes Tier 5 different: Each example frames a real-world decision problem and produces a concrete recommendation (e.g., "set contingency budget at the 90th percentile", "schedule maintenance before 95% failure probability").

Tier 6 — Cross-Domain Applications

Complex multi-physics examples combining techniques from earlier tiers.

# Example Domain Key Feature
16 Power Grid Frequency Regulation Electrical 4-coupled SDE (swing equations), two generators
17 Two-Compartment Drug Dosing Pharmacokinetics Coupled PK compartments, periodic dosing, therapeutic window
18 Solar Farm Battery Sizing Energy OU generation with diurnal solar cycle, battery clamping
19 Heat Exchanger Calibration Thermal Gradient optimization pipeline, inverse problem

What makes Tier 6 different: These examples combine multiple domains and techniques — coupled dynamics from Tier 4, optimization from Tier 5, and non-trivial physics (periodic forcing, clamped state, multi-generator coupling).

Example Walkthrough: RC Circuit Thermal Noise

The simplest engineering example demonstrates thermal noise in an RC circuit:

name: RC Circuit Thermal Noise
description: |
  Johnson-Nyquist noise in an RC low-pass filter.
  dV = -(V/RC) dt + sqrt(2 k_B T / C) dW

equation: "Y = sde($drift, $sigma, X)"

stochastic:
  calculus: ito
  n_paths: 1000
  dt: 0.001
  seed: 42

simulate:
  t_start: 0.0
  t_end: 0.1

initial_condition:
  x0: 0.0

params:
  drift: -100.0     # -1/RC, with RC = 10ms
  sigma: 0.0064     # sqrt(2 * k_B * T / C)

output:
  type: StochasticEnsemble
  title: "RC Circuit: Thermal Noise"

Physical interpretation: The voltage across the capacitor fluctuates around zero due to thermal agitation of charge carriers. The OU process captures the RC time constant (mean reversion) and the equilibrium voltage variance predicted by the fluctuation-dissipation theorem.

Next Steps