---
title: Extreme Learning Machines (ELM)
url: https://www.emergentmind.com/topics/extreme-learning-machines-elm
type: topic
---

# Extreme Learning Machines (ELM)

Operator Splitting Method (OSM) refers to a class of numerical methods for solving differential, variational, and monotone-inclusion problems by decomposing the original operator (or system) into a sum of simpler components, each of which is integrated or solved separately. This methodology is foundational across PDEs, DAEs, control, optimization, computational physics, and high-dimensional simulation, providing a rigorous structure for leveraging sparsity, structure preservation, and multirate effects.

## 1. Mathematical Foundations and Abstract Formulation

The prototypical operator splitting scenario involves a Cauchy problem or inclusion of the form
$$
\frac{du}{dt} = (A_1 + A_2 + \cdots + A_N)u,
$$
or, in monotone operator theory, find $x$ such that $0 \in A(x) + B(x)$ (or $A(x) + B(x) + C(x)$ in the three-operator case). Each $A_i$ is typically chosen to expose structure (e.g., linear, dissipative, Hamiltonian, constraint-enforcing) enabling specialized sub-solvers or analytic flows.

Splitting replaces the original flow $u(t) = e^{t(A_1+...+A_N)}u_0$ by compositions of flows $e^{\alpha_{i} t A_i}$ (with step weights $\alpha_i$), or by more general sub-integrators where exact exponentials are unavailable. The method’s order, stability, and structure preservation properties depend on the algebraic conditions on these coefficients, the sub-operator properties, and the chosen composition.

## 2. Classical Techniques: Lie–Trotter, Strang, and High-Order Schemes

The two-operator Lie–Trotter splitting advances the solution with sequential flows:
$$
u_{n+1} = e^{\Delta t A_2} e^{\Delta t A_1} u_n,
$$
yielding first-order global accuracy.

Strang (symmetric) splitting, 
$$
u_{n+1} = e^{\frac{\Delta t}{2}A_1} e^{\Delta t A_2} e^{\frac{\Delta t}{2}A_1} u_n,
$$
achieves second-order accuracy. The local error is controlled by the leading commutators in the Baker–Campbell–Hausdorff expansion.

For $N$-splitting ($N > 2$), Strang-type symmetric compositions generalize as
$$
S_h(A_1,\ldots,A_N) = e^{\frac{h}{2}A_1} \cdots e^{hA_N} \cdots e^{\frac{h}{2}A_1},
$$
with order conditions determined by commutator calculus. For arbitrary $N$ and higher order, recent complex-valued methods (e.g., CLT-2 for $N$-split) have been introduced, ensuring positive real weights in all fractional steps and favorable A-stability [2407.02677].

High-order splitting schemes are constructed by multi-stage compositions. Efficient third-order methods for two operators cannot avoid negative weights, and for higher order, optimizing stability and error constants (e.g., via the Local Error Measure—LEM) is crucial [2501.02310].

## 3. Structure Preservation: Energy, Dissipativity, and Symplecticity

In systems like linear port-Hamiltonian flows,
$$
\dot{x} = (J - R)\nabla H(x) + Bu,
$$
where $J$ is skew-symmetric, $R$ positive semi-definite, and $H$ quadratic, the splitting must preserve dissipativity:
$$
\frac{dH}{dt} = -\nabla H(x)^T R \nabla H(x) + y^T u \leq y^T u,
$$
and maintain the passivity structure [2406.17311].

A symmetric splitting (Strang or impulse methods), with energy-preserving integrators (e.g., implicit midpoint), ensures the discrete analog of this inequality, so the schemes do not artificially excite or dissipate energy inconsistent with the physical model.

For stochastic PDEs such as Maxwell's equations with additive noise, splitting into one-dimensional subsystems simultaneously preserves multi-symplectic laws and the correct growth rate of expected energy, under suitable regularity assumptions and careful numerical implementation [2102.10547].

## 4. Multirate, Scalar, and Distributed Splitting

Operator splitting methods leverage multirate potential by decoupling fast and slow subsystems, enabling macro-stepping for slow variables and micro-stepping for fast dynamics. This is particularly effective when coupled subsystems evolve on disparate timescales, such as mass-spring-damper chains with stiff and soft blocks [2406.17311].

Distributed operator splitting generalizes to large-scale monotone inclusion problems on networked graphs, where coefficient matrices encode local variables and inter-node communications, and fixed-point convergence is achieved via averaged or conically quasi-averaged operators. This yields highly scalable, decentralized algorithms applicable to consensus optimization, dual decomposition, and networked control [2504.14987].

## 5. Optimization, Monotone Inclusions, and Accelerated Splitting

Three-operator (Davis–Yin) splitting solves inclusions of the form $0 \in A x + B x + C x$ where $A, B$ are set-valued maximally monotone, and $C$ is single-valued and cocoercive:
```text
x_B = J_{γB}(z)
x_A = J_{γA}(2x_B - z - γC x_B)
z⁺  = z + λ(x_A - x_B)
```
Unification of classical Forward–Backward and Douglas–Rachford schemes arises as special cases via appropriate operator choices. In convex optimization, these splittings deliver near-optimal rates and permit acceleration (Nesterov-type) for strongly monotone scenarios [1504.01032], with recent extensions to inertial, line-search, and composite settings.

Variable and operator splitting (VOS) introduces independent auxiliary variables, transforming classic gradient flow into a coupled dynamical system. This enables strong Lyapunov functions, yielding explicit accelerated algorithms (AOR, EPC), and optimal complexity, including for minimax and saddle-point problems [2505.04065].

## 6. Applications in PDEs, DAEs, Control, and Simulation

OSM is essential in the efficient numerical solution of PDEs arising in finance (option pricing) via ADI, IMEX, and complementarity-projection splitting schemes [1504.01022], phase-field models (e.g., Cahn–Hilliard equations) with unconditional discrete energy stability [2107.01418], and nonlinear ODEs via separation of variables and Koopman–Lie semigroup techniques [2506.17524].

In DAEs, care must be taken to preserve algebraic constraints; splitting must respect the index-1 structure and energy/power conservation (e.g., ε-regularization in port-Hamiltonian DAEs) [2308.16736].

Efficient simulation of multi-physics systems such as dynamic gas flows in pipeline networks is achieved via explicit split-step methods leveraging the analytic flows of linear and nonlinear sub-operators, ensuring causality, mass-conservation, and robustness to network topology [1611.10008].

Operator splitting with capillary relaxation (OSCAR) separates viscous and capillary scales in two-phase flows, achieving dramatic speed-ups and controllable splitting error at low capillary number [2105.10576].

## 7. Practical Considerations: Cost, Stability, and Efficiency

The computational advantage of operator splitting methods emerges from four factors:
- Decomposition yields smaller linear systems or enables closed-form sub-flows.
- Multirate and scalar splitting reduces the dimensionality of computational blocks.
- Stability is optimized by proper ordering of operators and tailored sub-integrators, including replacement of unstable implicit steps by explicit schemes at minimal loss of formal accuracy [2501.02310].
- Block-iterative, asynchronous, or distributed implementations allow parallelization and scalability, critical for network control, large-scale optimization, and gray-box modeling [2310.08443], [2505.19411].

In multi-operator scenarios ($N > 2$), efficient compositions—e.g., complex-valued CLT-2 methods—permit favorable stability with positive real steps and serve as bases for constructing high-order, $N$-split schemes [2407.02677], [2302.08034].

Convergence rates are controlled by the sub-integrator order and the algebraic satisfaction of commutator-based constraints. Structure preservation and guaranteed monotonicity (e.g., dissipativity, passivity) in the discrete scheme are essential in physical models and optimization.

---

References:
- Operator splitting for coupled linear port-Hamiltonian systems [2406.17311]
- A pair of Second-order complex-valued, N-split operator-splitting methods [2407.02677]
- A new efficient operator splitting method for stochastic Maxwell equations [2102.10547]
- A Three-Operator Splitting Scheme and its Optimization Applications [1504.01032]
- Improving the stability and efficiency of high-order operator-splitting methods [2501.02310]
- A general approach to distributed operator splitting [2504.14987]
- Fast operator splitting methods for obstacle problems [2203.08380]
- Operator-Splitting Methods for Neuromorphic Circuit Simulation [2505.22363]
- Split-as-a-Pro: behavioral control via operator splitting and alternating projections [2505.19411]
- The Geometry of Monotone Operator Splitting Methods [2310.08443]
- Application of Operator Splitting Methods in Finance [1504.01022]
- The operator-splitting method for Cahn-Hilliard is stable [2107.01418]
- Operator Splitting Methods: Numerical Solutions of Ordinary Differential Equations via Separation of Variables [2506.17524]
- Operator splitting for semi-explicit differential-algebraic equations and port-Hamiltonian DAEs [2308.16736]
- GeoChemFoam: Operator Splitting based time-stepping for efficient Volume-Of-Fluid simulation of capillary-dominated two-phase flow [2105.10576]
- Accelerated Gradient Methods Through Variable and Operator Splitting [2505.04065]
- Operator Splitting Method for Simulation of Dynamic Flows in Natural Gas Pipeline Networks [1611.10008]
- Operator splitting for abstract Cauchy problems with dynamical boundary condition [2004.13503]
- Beyond Strang: A practical assessment of some second-order 3-splitting methods [2302.08034]

Source: https://www.emergentmind.com/topics/extreme-learning-machines-elm