EPDE Framework: Evolutionary PDE Discovery & Control

• EPDE framework is a suite of methods that use evolutionary optimization to recover the structure and coefficients of differential equations from spatio-temporal data.
• It leverages dynamic term generation and multi-objective symbolic regression to explore complex operator spaces and maintain noise robustness.
• EPDE extends to optimal control and fractional PDEs by reformulating problems into Lyapunov-inspired infinite-dimensional evolution equations for efficient computation.

The EPDE (Evolution Partial Differential Equation or Evolutionary PDE Discovery) framework designates a class of methodologies encompassing both evolutionary optimization-based discovery of governing equations from data and Lyapunov-inspired infinite-dimensional optimization schemes for optimal control. Multiple research lines and applications share the acronym, including data-driven PDE identification, variation-evolving methods for optimal control, and parametric reformulations for fractional differential equations.

1. Definitions and High-Level Overview

In data-driven equation discovery, "EPDE" refers to an evolutionary computation framework designed to flexibly recover the structure and coefficients of ordinary and partial differential equations governing spatio-temporal data fields, without reliance on a fixed pre-specified term library as in regression approaches such as SINDy. EPDE dynamically constructs and evolves candidate PDE forms via genetic operators and evaluates their fitness against observed data, allowing recovery of non-standard operators and greater noise robustness (Maslyaev et al., 2019, Maslyaev et al., 2024).

In the theory and computation of optimal control, "EPDE" denotes the Evolution Partial Differential Equation arising from the Variation Evolving Method (VEM). This constructs an infinite-dimensional evolution equation in an artificial "variation time" τ for trajectories and controls, with the property that its solution asymptotically approaches the minimum of a Lyapunov-type functional—typically the OCP cost—while rigorously maintaining feasibility and handling terminal equality/inequality constraints (Zhang et al., 2017, Zhang et al., 2018, Zhang et al., 2017, Zhang et al., 2018, Zhang et al., 2018).

A third context is the reduction of time-fractional differential equations (TFDEs) to coupled integer-order systems via parametric lifting, yielding an extended-parametric differential equation (EPDE) for efficient high-order computation (Ding et al., 10 May 2025).

2. EPDE for Data-Driven Equation Discovery

The EPDE discovery framework operates as an evolutionary-symbolic regression system, focused on reconstructing a governing PDE from field data $u(x, t)$ without substantial a priori structural assumptions. Its core workflow involves:

• Dynamic Term Generation: Rather than using a fixed term library, EPDE randomly generates terms by combining elementary differentiation, polynomial, and trigonometric tokens into products, thus exploring a broader search space (Maslyaev et al., 2024, Maslyaev et al., 2019).
• Candidate Encoding: Each individual (chromosome) is a candidate PDE comprising a small number of terms. Each term is built from tensor products of feature factors, such as normalized $u$, derivatives $\partial^k u/\partial x^{k_1}\partial t^{k_2}$, and potentially non-polynomial operators.
• Fitness Evaluation: Multiple modes are supported:
  • Discrepancy-based regression fitness, using a designated target term and weighted least-squares/LASSO regression;
  • Solver-based fitness, leveraging PINNs or finite-difference solvers to directly compare solution fields to data.

In both cases, a complexity penalty encourages parsimony (Maslyaev et al., 2024).

• Multi-objective Evolution: EPDE formulates the search as a multi-objective optimization problem, seeking to minimize both accuracy loss ($Q$) and complexity ($C$), and employs MOEA/D for Pareto front exploration.
• Genetic Operators: Reproduction applies crossover (at the level of terms and sub-terms), as well as mutation (adding, removing, or perturbing tokens), with mechanisms to control redundancy and ensure derivative-presence constraints (Maslyaev et al., 2024, Maslyaev et al., 2019).
• Output: The Pareto-optimal set of discovered equations exposes the accuracy/simplicity trade-off.

This approach enables the recovery of nonlinear, non-polynomial, and nonlocal operators, demonstrates higher noise robustness compared to fixed-library regression (notably at moderate noise levels), and maintains discovery capability where traditional SINDy fails (Maslyaev et al., 2024, Maslyaev et al., 2019).

Feature	EPDE (Discovery)	SINDy
Term Generation	Dynamic, evolutionary	Fixed library
Optimization	Multi-objective, evolutionary	LASSO/sparse regression
Noise Robustness	High (with PINN/ANN preproc)	Degrades quickly
Discovered Forms	Arbitrary (including novel)	Limited by library
Compute Cost	High (10³–10⁴× SINDy)	Low

3. EPDE in Optimal Control via Variation Evolving Method

The VEM-based EPDE formulates the solution of Bolza-type optimal control problems as the asymptotic equilibrium of an evolution equation in a virtual ("variation") time τ. The essential elements include:

• Lyapunov Framework: The cost functional $J[x, u, t_f]$ is regarded as a Lyapunov functional on the space of admissible trajectories, and τ-dynamics are constructed such that $dJ/d\tau \leq 0$ with equality only at the optimum (Zhang et al., 2017, Zhang et al., 2018).
• EPDE Structure: The coupled evolution equations for states $x(t, \tau)$, controls $u(t, \tau)$, and occasionally multipliers or terminal times, take the general form

$$\frac{\partial u(t, \tau)}{\partial \tau} = -K \nabla_u J$$

where the gradient may involve integrals over future time, reflecting costate-free, noncausal dependencies in general nonlinear OCPs.

• Costate-free Optimality: The stationarity conditions at τ-equilibrium recapitulate the Pontryagin minimum principle, yet are expressed in the primal state/control variables (and analytic multipliers) without requiring explicit adjoint integration (Zhang et al., 2017, Zhang et al., 2018).
• Constraint Handling: Terminal equality and inequality constraints are incorporated by suitable algebraic or soft-barrierized multipliers, enabling rigorous handling within the evolution framework (Zhang et al., 2018, Zhang et al., 2018).
• Numerical Integration: The method of lines discretizes the physical time domain, yielding a high-dimensional ODE initial-value problem (IVP) in τ that is tractable by standard ODE integrators (e.g., MATLAB ode45, ode15s), with monotonic descent guarantees.

Variants have introduced compact control-only evolutions (reducing state dimensionality by reconstructing states and adjoints algebraically at each τ-step) (Zhang et al., 2018), arbitrary-definite-condition EPDEs that relax the feasible-initial-guess requirement via an unconstrained Lyapunov functional (Zhang et al., 2017), and efficient semi-discretizations for dense grids.

EPDE Variant (Optimal Control)	Unknowns Evolved in τ	Handles Infeasible Start?	Costate Needed?
Primal VEM EPDE	$x$, $u$, $t_f$	Sometimes	No (recovered)
Compact (control-only) EPDE	$u$ (states/costates by quadrature)	Yes	No (algebraic)
Arbitrary-definite EPDE	$x$, $u$, $t_f$ plus errors	Yes	No

4. EPDE for Fractional Differential Equations: Parametric Reformulation

In the context of time-fractional (Caputo-derivative) PDEs, the EPDE construction "lifts" the nonlocal-in-time fractional operator to a local-in-time, integer-order parametric system via an auxiliary variable $\theta$:

• Parametric Lifting: The Caputo derivative is represented as an integral over exponentially decaying kernels parameterized by $\theta \in (0,1)$, resulting in an unknown $U(x, t, \theta)$ satisfying an integer-order PDE in $(x, t, \theta)$ (Ding et al., 10 May 2025).
• Numerical Discretization: The extended PDE is discretized via Jacobi spectral collocation in $\theta$ and arbitrary high-order BDF-$k$ in $t$. The original solution $u(x, t)$ is recovered by quadrature over $\theta$.
• Efficiency: This formulation yields optimal $O(N)$ cost and $O(1)$ storage (for $N$ time-steps), as opposed to $O(N^2)$ work and $O(N)$ memory in direct approaches due to the mitigation of history dependence.

5. Comparative Performance and Applications

Experimental investigations demonstrate:

• Equation Discovery: EPDE achieves near-perfect recovery on canonical PDEs (Burgers, KdV, wave) in noise-free settings, and maintains non-trivial discovery rates and low coefficient error (MAPE, RMSE) up to moderate superposed noise levels, exceeding baseline regression approaches (Maslyaev et al., 2019, Maslyaev et al., 2024).
• Optimal Control: In both linear-quadratic and nonlinear OCPs, EPDE frameworks converge reliably from arbitrary initializations, yielding trajectories and controls matching direct/collocation solvers. Compact control-only EPDEs excel in accuracy and speed for dense discretizations (Zhang et al., 2017, Zhang et al., 2018).
• Fractional Equations: EPDE-based numerical schemes for TFDEs exhibit spectral-convergent accuracy in the auxiliary parameter, high-order time convergence, and efficiency matching integer-order solvers irrespective of fractional order (Ding et al., 10 May 2025).

6. Limitations, Extensions, and Research Directions

Limitations observed across EPDE methodologies include:

• Computational Burden: Evolutionary discovery variants are orders of magnitude slower than regression-based methods, with many hyperparameters requiring tuning (Maslyaev et al., 2024).
• Repeatability: Stochastic search (evolutionary discovery) and random initialization induce irreproducibility unless carefully ensemble-averaged or seed-controlled (Maslyaev et al., 2019).
• Data Requirements: Both discovery and control EPDEs require sufficiently rich data coverage or physical domain discretization; performance degrades when data or grid is undersampled (Maslyaev et al., 2019, Zhang et al., 2017).
• Noise Sensitivity: While more robust than standard methods, correct structure recovery from high-noise data remains challenging, even with PINN-based fitness and advanced preprocessing (Maslyaev et al., 2024).

Ongoing and potential extensions include adaptive operator grammar, surrogate-model-based fitness, integration of physical priors (invariants, symmetries), advanced genetic operators (adaptive mutation, island models), and incorporation into real-world experimental pipelines (Maslyaev et al., 2019, Maslyaev et al., 2024).

7. Concluding Synthesis

The umbrella term "EPDE framework" encompasses a suite of powerful techniques centered on infinite-dimensional evolutionary or evolution-inspired approaches for discovering and optimizing operators within partial differential, ordinary differential, and fractional systems. In both data-driven and control-theoretic regimes, EPDE methods provide principled, flexible, and theoretically justified procedures with demonstrated empirical efficacy—at the expense of higher computational overhead and parameter complexity compared to conventional methods. Further research is warranted into scaling, automation of hyperparameter tuning, and application to high-noise, high-dimensional real datasets (Maslyaev et al., 2024, Zhang et al., 2017, Zhang et al., 2018, Ding et al., 10 May 2025).