DeePoly - High-Order Accuracy Hybrid Neural Network Framework

Introduction

DeePoly is a universal high-precision mesh-free PDE solving framework. The core concept leverages the nonlinear approximation capabilities of deep neural networks combined with the local high-precision capabilities of traditional polynomial approximation. This novel approach significantly improves the accuracy and convergence limitations of PINN-based methods while addressing the efficiency disadvantages of non-convex optimization, achieving high-precision, high-efficiency universal equation solving capabilities. The framework aims to extend AI4S methods to complex equations and complex engineering problem solving.

Key Features

• Universal PDE Solver: Applicable to all kinds of PDEs
• Mesh-Free Approach: Sampling points can be randomly generated with no logical relationships, suitable for complex geometries
• High Accuracy: Achieves high-order convergence
• Scheme-Free: Handles derivative relationships using automatic differentiation
• Auto Code Generation: Write equations into config.json with no need to change any code
• Computational Efficiency: Performance comparable to traditional finite difference methods
• GPU Acceleration: Supports CPU parallelism and GPU acceleration
• Complex and Discontinuous Problems: Accurately approximates discontinuous and high-gradient functions (preliminary testing)
• Inverse Problems: Solves inverse problems with higher accuracy than PINNs (under development)

Research & Publications: ResearchGate | arXiv

New Version Highlights

Enhanced Nonlinear Equation Solving

Equation Form: $$\frac{\partial U}{\partial t} + L_1(U)+ L_2(U)F(U)+N(U)=0$$

Current Time Integration: Second-order implicit time scheme implemented in onestep_predictor.py (currently the only recommended time format).

Performance Highlights:

• Allen-Cahn Equation: Demonstrates second-order temporal accuracy and high efficiency
• Fast Computation: AC_equation_100_0.1 case completes in only 10 seconds
• Tested Applications: Allen-Cahn, KdV, and Burgers equations successfully validated
• Future Development: Navier-Stokes equations and other 2D multi-equation systems in progress

Computational Results:

Allen-Cahn equation solution at final time with only 100 fully random points and dt=0.1 large time step

Unified Codebase Maintenance & Organization

• Unified Configuration System: BasePDEConfig provides consistent eq={L1,L2,F,N,S} semantics across all solvers
• Factory Pattern Architecture: Single entry point with consistent solver selection and workflow management
• Dimension Standardization: Consistent tensor operations and shape management throughout the codebase

Boundary Condition Unification

• Unified Processing: Consistent boundary condition specification and handling across all solver types
• Simplified Periodic Boundaries: Unified paired constraint implementation replacing complex multi-type systems

• Boundary Condition Types

Unified boundary condition processing supports:

• Dirichlet: "type": "dirichlet" with value specification
• Neumann: "type": "neumann" with normal derivative values
• Robin: "type": "robin" with mixed boundary conditions
• Periodic: "type": "periodic" with automatic pairing

🔧 Intelligent Code Generation & Maintenance

• Automatic Code Generation: auto_spotter.py automatically generates/updates PDE-specific code segments
• Consistency Verification: Signature checking and automatic regeneration of outdated code
• Configuration-Driven: Direct code generation from config.json specifications

Quick Start

Installation Requirements

# Core dependencies
pip install numpy scipy torch matplotlib

# Optional GPU acceleration
pip install cupy-cuda12x  # For CUDA 12.x

# Development environment
conda activate your_ml_environment

Basic Usage

Time PDE (Allen-Cahn Equation)

python src/main_solver.py --case_path cases/Time_pde_cases/Allen_Cahn/AC_equation_100_0.1

Linear PDE (Poisson 2D)

python src/main_solver.py --case_path cases/linear_pde_cases/poisson_2d_sinpixsinpiy

Function Fitting (2D Complex Function)

python src/main_solver.py --case_path cases/func_fitting_cases/case_2d

Configuration Format

Unified Configuration System

DeePoly uses a standardized eq={L1,L2,F,N,S} format across all problem types:

Linear PDE Example (Poisson)

{
  "problem_type": "linear_pde",
  "eq": {
    "L1": ["diff(u,x,2) + diff(u,y,2)"],
    "L2": ["0"],
    "F": ["0"],
    "N": ["0"],
    "S": ["-sin(pi*x)*sin(pi*y)"]
  },
  "vars_list": ["u"],
  "spatial_vars": ["x", "y"],
  "x_domain": [[0,1], [0,1]],
  "n_segments": [1, 1],
  "poly_degree": [10, 10],
  "boundary_conditions": [
    {
      "type": "dirichlet",
      "region": "left",
      "value": "1/2/(pi)**2*sin(pi*x)*sin(pi*y)",
      "points": 100
    }
  ],
  "auto_code": false
}

Time PDE Example (KdV)

{
  "problem_type": "time_pde",
  "eq": {
    "L1": ["1.0*diff(u,x,3)"],     // Implicit stiff terms
    "L2": ["diff(u,x)"],           // Semi-implicit linear terms
    "F": ["u"],                    // Functions for L2⊙F coupling
    "N": []                        // Explicit nonlinear terms
  },
  "vars_list": ["u"],
  "spatial_vars": ["x"],
  "x_domain": [[-20, 20]],
  "T": 40.0,
  "dt": 0.1,
  "time_scheme": "onestep_predictor",
  "Initial_conditions": [
    {"var": "u", "value": "cos(pi*x/20)", "points": 1000}
  ],
  "boundary_conditions": [
    {"type": "periodic", "region": "left", "pair_with": "right", "points": 1}
  ],
  "auto_code": false
}

Source Term and Reference Solution Support

DeePoly supports multiple formats for source terms and reference solutions:

{
  "eq": {
    "S": ["sin(pi*x)*cos(pi*y)"]           // Mathematical expression
    // OR
    "S": "data_generate.py"                 // Python file reference
  },
  "reference_solution": "analytical_solution",     // Mathematical expression
  // OR
  "reference_solution": "data_generate.py",        // Python function
  // OR
  "reference_solution": "reference_data.mat"       // MATLAB data file
}

Available Examples

Linear PDE Cases

• poisson_2d_sinpixsinpiy: 2D Poisson equation with analytical solution
• test_file_source: Demonstrates source term loading from files

Time PDE Cases

Allen-Cahn Equation (Allen_Cahn/AC_equation_100_0.1)

Allen-Cahn equation with second-order time integration - recommended example showing fast 10-second computation

Other Time PDE Examples

• KDV_equation: Korteweg-de Vries equation
• Burgers: Burgers equation

Note: All time PDE examples use the second-order predictor-corrector time integration scheme.

Function Fitting Cases

• case_2d: Complex 2D multi-modal function approximation
• discontinuous_case1: Challenging discontinuous function handling

Version History

• v0.1 (Beta): Initial release with basic function approximation and PDE solving
• v0.2: High-accuracy linear PDE solving, auto-code generation, English documentation
• v0.3 (Current): Enhanced nonlinear solving, unified system architecture, advanced time integration, comprehensive boundary condition unification

Next Version Roadmap

• 2D Time-Dependent Nonlinear PDEs: Complete implementation of two-dimensional time-dependent nonlinear PDE problem solving and steady-state problems
• Pseudo-Time Stepping: Global solving with pseudo-time stepping (removing time scheme precision limitations on solution accuracy)

Future Development

• Complex Boundary Conditions: Advanced internal and external boundary conditions (leveraging mesh-free method advantages)
• Discontinuous Problem Enhancement: Improved accuracy for discontinuous problems and compressible fluid dynamics problem solving。

Citation

If you use DeePoly in your research, please cite:

@article{liu2025deepoly, title={DeePoly: A High-Order Accuracy Scientific Machine Learning Framework for Function Approximation and Solving PDEs}, author={Liu, Li and Yong, Heng}, year={2025} }

Test Cases & Validation Results

The following test cases demonstrate DeePoly's capabilities across function approximation and PDE solving:

Function Approximation Cases

1. 1D Sine Function

Case Directory: func_fitting_cases/test_sin Target Function: f(x) = sin(2πx) + 0.5cos(4πx) Problem Type: Baseline function approximation test case

Results

Training Data Analysis

Test Data Analysis

Performance Metrics

| Component | Time Cost | |-----------|-----------| | Scoper/DNN | 7.2049 seconds | | Sniper | 0.1630 seconds | | Total | 7.3679 seconds |

Additional Information

• Error Analysis: Detailed metrics available in cases/func_fitting_cases/test_sin/results/error_analysis_report.txt
• Configuration: See cases/func_fitting_cases/test_sin/config.json

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2. Complex 2D Function

Case Directory: func_fitting_cases/case_2d Target Function: f(x,y) = Multi-Gaussian peaks + trigonometric + polynomial terms on [-3,3]² Problem Type: Complex multi-modal function approximation

Results

Training Data Analysis ![Training Results](cases/func_fitting_cases/