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Overview

What is Reservoir Computing?

Reservoir Computing (RC) is a computational framework for processing temporal and sequential data. Unlike traditional recurrent neural networks that require expensive backpropagation through time, RC keeps the recurrent layer (the reservoir) fixed and only trains a simple output layer (the readout). This makes training fast, often reducing to a single ridge regression solve, while still capturing complex temporal dynamics.

RC is particularly effective for:

  • Chaotic system forecasting — predicting trajectories of systems like the Lorenz attractor
  • Spatiotemporal modeling — forecasting PDEs like the Kuramoto-Sivashinsky equation
  • Time series prediction — general sequential data analysis
  • System identification and control — learning input-output relationships of dynamical systems

Core Architecture

ORC decomposes every reservoir computing model into three modular layers:

Input → [Embedding] → [Driver] → [Readout] → Output

Embedding Layer

The embedding layer maps input signals from the data dimension into the reservoir dimension. It defines how external data enters the reservoir.

  • ParallelLinearEmbedding — random linear projection with support for spatial decomposition via overlapping chunks
  • LinearEmbedding — single-reservoir linear embedding
  • EnsembleLinearEmbedding — multiple independent embeddings for ensemble models

See Embeddings for details.

Driver Layer

The driver layer implements the reservoir's temporal dynamics. It takes embedded input and the current reservoir state, then computes the next state.

  • ParallelESNDriver / ESNDriver — Echo State Network dynamics with tanh nonlinearity, supporting discrete and continuous time modes
  • ParallelTaylorDriver / TaylorDriver — Taylor-expanded ESN dynamics for computational efficiency
  • GRUDriver — Gated Recurrent Unit dynamics

See Drivers for details.

Readout Layer

The readout layer is the trainable component. It maps reservoir states to desired outputs, typically trained via ridge regression.

  • ParallelLinearReadout / LinearReadout — standard linear mapping
  • ParallelNonlinearReadout / NonlinearReadout — user-defined nonlinear transformations before linear mapping
  • ParallelQuadraticReadout / QuadraticReadout — quadratic feature expansion
  • EnsembleLinearReadout — ensemble averaging across multiple readouts

See Readouts for details.

Available Models

ORC provides ready-to-use models that combine these layers for common tasks:

Forecasting

  • ESNForecaster — discrete-time Echo State Network for autonomous prediction
  • CESNForecaster — continuous-time ESN using Diffrax ODE solvers
  • EnsembleESNForecaster — multiple independent ESNs with averaged predictions

Classification

  • ESNClassifier — time series classification using reservoir features

Control

  • ESNController — system control via reservoir-based state estimation

See Models for usage details.

Training

ORC models are trained using ridge regression (Tikhonov regularization), which solves a linear system rather than iterating with gradient descent. The typical workflow is:

  1. Force the reservoir with training data to collect reservoir states
  2. Solve the ridge regression problem to find readout weights
  3. Forecast autonomously from the final reservoir state
import orc

# Create and train
esn = orc.forecaster.ESNForecaster(data_dim=3, res_dim=500, seed=42)
esn, res_states = orc.forecaster.train_ESNForecaster(esn, U_train)

# Predict
U_pred = esn.forecast(fcast_len=1000, res_state=res_states[-1])

Parallel Reservoirs

For high-dimensional or spatiotemporal data, ORC supports parallel reservoir architectures. Instead of one large reservoir, the system uses multiple smaller reservoirs that each process a local region of the input with configurable overlap (locality) and boundary conditions (periodic).

This approach reduces computational cost from \(O(N^2)\) to \(O(N)\) in the spatial dimension while preserving local structure.

New users should start with the Quick Start tutorial, then read the component guides:

  1. Embeddings — input-to-reservoir mapping
  2. Drivers — reservoir dynamics
  3. Readouts — reservoir-to-output mapping
  4. Models — complete model implementations
  5. Data — built-in dynamical systems for benchmarking

The API Reference provides complete technical documentation, and the Examples demonstrate practical applications.