ODE Template#

The ODE (Ordinary Differential Equation) template creates neural network models that learn the underlying differential equations governing a dynamic system. These models are ideal for physical systems with continuous dynamics.

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Overview#

ODE models learn to represent systems of the form:

\[ \begin{align}\begin{aligned}\frac{dx}{dt} = f(x, u, t)\\y = g(x, u)\end{aligned}\end{align} \]

where:

  • \(x\) is the internal state vector

  • \(u\) is the input (excitation) vector

  • \(y\) is the output vector

  • \(f\) and \(g\) are learned functions

Key capabilities:

  • Automatically discovers hidden states needed to model the dynamics

  • Supports steady-state constraints for physically meaningful behavior

  • Allows initialization from known states for accurate simulation

  • Exports to FMU format for integration with simulation tools

When to Use ODE#

The ODE template is ideal for:

  • Physical systems: Mechanical, thermal, electrical, hydraulic systems

  • Continuous dynamics: Systems governed by differential equations

  • State-based modeling: When the system has well-defined physical states

  • Long-term simulation: When prediction accuracy over extended time is critical

  • Digital twins: Creating virtual replicas of real-world systems

Example applications:

  • Mass-spring-damper systems

  • Heat transfer and thermal dynamics

  • Electric motor modeling

  • Vehicle dynamics

  • Process control systems

Model Architecture#

NeurEco ODE models consist of:

  1. State network: Learns the state dynamics \(f(x, u, t)\)

  2. Output network: Maps states to outputs \(g(x, u)\)

  3. Hidden states: Internal variables discovered during training

The number of hidden states is determined automatically during training, starting from min_hidden_states and growing as needed (up to max_hidden_states if specified).

Native File Format#

ODE models use the .ernn extension:

  • Preserves complete model information

  • Supports loading and continued training

  • Can be exported to FMU format

Workflow#

  1. Prepare data: Format time-series data with time, inputs, and outputs

  2. Build model: Train the ODE model on your data

  3. Evaluate: Run simulations with optional state initialization

  4. Export: Save in native format or export to FMU

ODE Template