PINN Soil Water Model via Automatic Differentiation

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Solution Overview

Problem

Existing methods for simulating soil water movement, such as those based on the Richards equation, face challenges in generality and achieving a balance between calculation speed and accuracy.

Innovation Solution

A method and system using Physical Informed Neural Networks (PINNs) to construct a soil water movement model, which involves constructing a partial differential equation, using a feedforward neural network to output an approximate solution, and employing automatic differentiation to obtain a residual network with a minimized loss function.

Engineering Contradictions & Design Principles

VSEngineering Contradiction Analysis

1Manufacturing precision

If traditional numerical methods (finite difference, finite element, finite volume) are used to solve the Richards equation, then the partial differential equation can be reduced to a finite-dimensional approximation problem, but the calculation accuracy is compromised due to spatial-temporal discretization errors and the method lacks generality

Engineering Contradiction:

Improvecalculation accuracyVSAvoidgenerality of calculation method

Core Design Contradiction:

Manufacturing precisionVSAdaptability or versatility

Solution Approach 1:

The patent replaces traditional mechanical numerical methods (finite difference, finite element, finite volume) with a neural network-based approach. The neural network learns the solution mapping from boundary and initial conditions to the solution field, eliminating the need for spatial-temporal discretization and associated numerical errors, thereby achieving both high accuracy and generality

Inventive Principle:

Principle #28Mechanics substitution (Replace mechanical system)

Solution Approach 2:

The patent transforms the problem from solving partial differential equations with fixed discretization parameters to a neural network learning problem where the network parameters (weights and biases) are optimized to minimize the loss function. This parameter transformation enables the model to achieve high accuracy without being constrained by discretization schemes

Inventive Principle:

Principle #35Parameter changes

2Productivity

If deep learning algorithms are used to create a surrogate model of hydrological processes, then calculation speed can be improved, but a very large number of data samples are required which are expensive or infeasible to obtain

Engineering Contradiction:

Improvecalculation speedVSAvoidnumber of training data samples

Core Design Contradiction:

ProductivityVSQuantity of substance

Solution Approach 1:

The patent embeds the Richards equation and its physical constraints into the neural network training process beforehand. By incorporating the governing PDE as a soft constraint in the loss function, the network learns physics-consistent solutions from minimal data, eliminating the need for large datasets while maintaining fast prediction speed

Inventive Principle:

Principle #10Preliminary action

Solution Approach 2:

The patent introduces physics constraints (the Richards equation) as an intermediary between the neural network and the training data. This intermediary guides the network to learn physically meaningful patterns from limited data, bridging the gap between data-driven speed and physics-based accuracy without requiring extensive training samples

Inventive Principle:

Principle #24Intermediary (Mediator)

3Stability of the object's composition

If the backward Euler scheme is used for time discretization of the Richards equation, then stability condition is improved, but the method still suffers from discretization errors and poor adaptability

Engineering Contradiction:

Improvestability conditionVSAvoidcalculation accuracy

Core Design Contradiction:

Stability of the object's compositionVSManufacturing precision

Solution Approach 1:

The patent replaces the backward Euler time discretization scheme with a neural network-based continuous solution approach. The neural network naturally handles time continuity without discretization, eliminating both the stability constraints of explicit schemes and the discretization errors of implicit schemes while maintaining accuracy

Inventive Principle:

Principle #28Mechanics substitution (Replace mechanical system)

Data Source

PatentUS20250181801A1Method and system for constructing soil water movement model based on physical information neural networks

Publication Date: 2025.06.05 CHINA INST OF WATER RESOURCES & HYDROPOWER RES

AI summary

A method and a system for constructing a soil water movement model based on the physical information neural networks are provided. The soil water movement model is constructed by fusing the physical information neural network and a control equation of the soil water movement model, and the model adopts automatic differentiation to replace difference operation of grid scale, so that the calculation error caused by equation discretization in the solving process of numerical differentiation is avoided, and the calculation accuracy is improved; and the automatic differentiation mode is mainly carried out aiming at the output of a neural network, so that the numerical value of gradient calculation is more accurate.