Modified Partial Differential Equations (PDEs)

A modified partial differential equation (PDE) refers to any partial differential equation that departs from a standard or canonical form—either by changes in its structure, the introduction of new terms, altered coefficients, time- or space-dependent geometry, or coupling to external or evolving systems. The concept of modified PDEs encompasses both analytic modifications made in mathematical analysis and those emerging experimentally or numerically, and is a central theme in modern applied mathematics, mathematical physics, and computational science.

1. General Characterization of Modified PDEs

A modified PDE can generally be expressed as: $$\mathcal{F}(u, \nabla u, \nabla^2 u, \ldots; \lambda, x, t) = 0,$$ where $\mathcal{F}$ is not fixed but subject to systematic or data-driven alterations. Modifications can include the addition of source or sink terms (e.g., cubic sources), changes to spatial or temporal operators, the inclusion of integral or nonlocal terms, space- or solution-dependent coefficients, or coupling to other equations or moving domains. Modified PDEs thus form a broad class, extending classical models such as the Korteweg-de Vries or Burgers equations to far richer dynamical regimes.

Modified PDEs in Theory and Applications

• Theory: They are used to uncover or classify new exact solutions, refine models by including neglected physical effects, or as auxiliary tools for analytical or numerical studies.
• Applications: Modified PDEs are prevalent in modeling real-world phenomena where canonical models are inadequate—for instance, nonlinear waves in multiphase media, morphogenesis in developmental biology, or advanced image processing in hazy or underwater conditions.

2. Analytical and Algebraic Methods for Modified PDEs

Numerous analytical frameworks have been developed to construct, analyze, or solve modified PDEs:

Modified Simplest Equation Method

This approach expresses the solution to a nonlinear PDE in terms of solutions to a simpler, typically linear, auxiliary equation. The process includes:

1. Singularity Analysis: Determine pole/zero structure by dominant balancing.
2. Solution Ansatz: Postulate $u(x, t)$ as a function (e.g., rational, polynomial, logarithmic) of the auxiliary solution $Z$, such as

$u(x, t) = a_0 + a_1 \frac{Z_x}{Z}$

for first-order poles, or

$u(x, t) = a_0 + a_1 Z + \dots + a_n Z^n$

for zeros.

1. Reduction and Compatibility: Substitute into the original PDE, reduce higher derivatives using the auxiliary equation, and solve the resulting algebraic system for parameters and coefficients.
2. Construction of Exact Solutions: Solve the reduced (typically linear) equation for $Z$ to reconstruct $u(x, t)$.

This method enhances traditional integration techniques by accommodating a broader class of nonlinear and non-traveling solutions and is illustrated by exact analytic solutions to generalized KdV equations with cubic source terms and third-order Kudryashov–Sinelshchikov equations that model nonlinear waves in bubbly fluids.

Transformation and Operator Approaches

Other analytical frameworks construct modified PDEs through systematic transformation. For example:

• Partial Integral Equation (PIE) Representation: Linear PDEs (even with integral terms in the dynamics and boundaries) are converted via explicit operator maps into PIEs, eliminating the need for explicit boundary conditions and enabling analysis in $L_{2}$ spaces using operator-theoretic and convex optimization tools. This mapping is exact and invertible under appropriate admissibility conditions.
• Modified Evolution PDEs for Optimization (MEPDE): In optimal control, PDEs are modified to embed artificial evolution in "variation time" $\tau$, allowing solutions to ordinary or boundary-infeasible problems to converge to feasible and optimal states through a Lyapunov-based stabilization process.

3. Data-Driven Discovery of Modified PDEs

In contemporary practice, especially with abundant measurement or simulation data, neural and evolutionary computation methods are used to discover modified PDEs that best fit observed phenomena:

Symbolic and Hybrid Machine Learning Methods

• Symbolic Genetic Algorithms (SGA-PDE): PDE terms are represented as forests of binary trees, and genetic algorithms evolve both structure and coefficients to identify open-form PDEs from data. This allows the automatic discovery of highly nontrivial terms, including nonlinear, fractional, and compound function PDEs that go beyond fixed candidate dictionaries.
• Numeric-Symbolic Hybrid Networks (PDE-Net 2.0): Convolutional filters approximate differential operators, while a symbolic multi-layer neural network ("SymNet") infers the nonlinear response function. This hybrid learns both the operators and the PDE structure, providing interpretable, data-driven discovery.
• Bayesian Sparse Regression Frameworks: Variational Bayes with spike-and-slab priors enables robust PDE discovery even under noise by selecting the most relevant terms from a comprehensive candidate dictionary, including nonlinearities and higher-order derivatives. This facilitates the discovery of not only standard equations but their modifications as required by data.

Practical Examples

Numerical experiments have demonstrated the recovery of modified PDEs such as:

• Generalized KdV equations of the form:

$$u_t + 3u^2u_x + 3(uu_x)_x + u^3 + u_{xxx} + p_1 u(u^2 + p_2 u + p_3) + p_4 u_x + p_5 u_{xx} + p_6 u u_x = 0,$$

where coefficients $p_i$ allow systematic exploration of physical regimes.

• PDEs for traffic dynamics that are higher-order, nonlinear and data-adaptive, identified from real-world sensor data using neural networks and sparse regression.

4. Modified PDEs in Complex and Evolving Domains

Recent frameworks extend the concept of modification to PDEs defined on non-static, spatially evolving domains or in conjunction with hybrid logical systems:

• Developmental PDEs (DPDEs): PDEs are coupled on time-varying manifolds, with mutual feedback between the solution and domain evolution. For example, in biological morphogenesis, a diffusion equation on a growing surface is fully coupled to the geometry, with controllability of the growth shape via source terms.
• Partial Differential Hybrid Automata (PDHA): Hybrid automata with discrete logical switching are extended to infinite-dimensional state spaces governed by PDEs, with spatial partitions enabling the modeling of spatially-localized discrete-continuous behavior. This approach supports, for example, traffic flow with localized congestion transitions, or heater control with zone-specific switching, by assigning different PDEs to spatial partitions with dynamic updates.

5. Approximation and Numerical Methods for Modified PDEs

Many practical scenarios require mapping large, potentially high-dimensional systems to tractable PDEs:

• PDE Approximation of ODE Systems: Large tridiagonal (birth-death-type) ODE systems can be approximated by parabolic PDEs with modified (dynamic or Robin/Wentzell) boundary conditions. The approximation ensures analytic tractability, accelerates computation (e.g., via spectral/Fourier methods), and captures conservation properties.
• Neural Generative and Mesh-Free PDE Solvers: Modern diffusion models (e.g., DiffusionPDE) model the joint solution–coefficient distribution, enabling the simultaneous recovery of both under sparse/broken measurement scenarios and supporting both forward and inverse PDE tasks. Pretrained foundation models (e.g., PDEformer-1), equipped with graph-based and mesh-free neural representations, further allow rapid adaptation to novel or modified one-dimensional PDEs in symbolic and numerical regimes.

6. Applications and Impact

Modified PDEs are essential in diverse applications:

• Physics and Engineering: Multi-phase fluid dynamics, elasticity with complex materials, macro- and micro-scale traffic modeling, and industrial image enhancement (haze, underwater, dust) all rely on modified PDEs for accurate representation, control, and signal recovery.
• Biology: Tissue morphogenesis, protein signaling on evolving domains, and other developmentally dynamic processes are described by coupled, domain-evolving PDEs.
• Cyber-physical Systems and Control: Hybrid automaton models augmented with PDE dynamics facilitate the control and analysis of spatially-distributed systems with both discrete and continuous transitions.
• Scientific Discovery: Automated, data-driven discovery techniques—combining symbolic, Bayesian, and neural learning—enable the identification, validation, and refinement of governing equations as experimental or simulated data demand new model forms.

7. Summary Table: Methods and Roles in Modified PDEs

The emergence, formulation, and computational treatment of modified PDEs exemplify both the adaptive evolution of mathematical modeling and the increasing integration of data-driven and symbolic computational paradigms in the analysis of complex, real-world systems.