Mechanism discovery method and system based on reinforcement learning inspired genetic programming
By constructing a dynamic mechanism sub-expression pool and performing mechanism-guided initialization using a reinforcement learning-inspired genetic programming method, the problem of extracting explicit physical equations of complex nonlinear systems in existing technologies is solved, and efficient and interpretable dynamic law extraction and control are achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- BEIHANG UNIV
- Filing Date
- 2026-05-19
- Publication Date
- 2026-07-31
AI Technical Summary
Existing technologies struggle to automatically extract explicit physical equations from complex nonlinear systems in multidimensional time-series observation data. Deep learning black-box models cannot extract physical mechanisms. Physical information neural networks rely on known equations. Traditional genetic programming is prone to destroying high-value physical building blocks. Deep reinforcement learning training is unstable and computationally expensive.
We employ a reinforcement learning-inspired genetic programming approach. By constructing a dynamic pool of reusable and novel physical mechanism sub-expressions, and combining mechanism-guided initialization with genetic programming evolutionary search, we extract and reuse high-value physical substructures across stages, and output explicit analytical equations.
It enables clear characterization of causal relationships between physical quantities from multidimensional time-series observation data, meets the needs of engineering and scientific fields for physical consistency, improves convergence efficiency and interpretability, and reduces engineering costs.
Smart Images
Figure CN122491394A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of nonlinear system dynamics mechanism discovery, and more specifically to a mechanism discovery method and system based on reinforcement learning-inspired genetic programming. Background Technology
[0002] In cutting-edge scientific and engineering fields such as aerospace, fluid mechanics, plasma physics, and chemical reaction dynamics, the spatiotemporal evolution of complex nonlinear systems is usually governed by a set of unknown partial differential equations or ordinary differential equations. Accurately obtaining the above-mentioned dynamic mechanism analytical equations is the core foundation for realizing tasks such as predicting the aerodynamic and thermal environment of hypersonic vehicles, plasma confinement control of tokamak devices, and optimization of chemical reaction processes. However, with the increasing degree of nonlinear coupling of systems and the surge in the dimensionality of observation data, traditional mathematical derivation methods relying on first principles are no longer sufficient to meet the complex physical field evolution modeling needs in practical engineering.
[0003] For high-dimensional nonlinear systems, existing pure data-driven models mainly use recurrent neural networks or transformer architectures for end-to-end learning. Although such methods can approximate the input-output mapping relationship of the system to a certain extent, the hundreds of millions of weight parameters inside the network completely lack the physical meaning at the mathematical analytical level and cannot output explicit control equations with algebraic structure. More importantly, black-box models cannot guarantee basic physical consistency such as energy conservation and momentum conservation, and are prone to prediction divergence under extreme conditions outside the distribution range of training samples. This makes it difficult to meet the practical engineering needs of aerospace vehicle control and nuclear fusion device regulation, which have strict requirements for interpretability and safety.
[0004] To compensate for the lack of physical consistency in purely data-driven models, researchers have proposed physical information neural networks. This method constrains the solution space of the model by transforming known physical laws into residual terms of partial differential equations and embedding them into a loss function. However, the essence of physical information neural networks is to solve inverse problems using existing equations, which requires providing complete system control equation expressions before modeling. When faced with complex dynamic systems containing unknown perturbations or entirely new physical mechanisms, the lack of prior equation forms prevents the construction of effective physical loss terms, thus fundamentally lacking the ability to proactively discover unknown mechanisms from scratch. This limits its application in real-world scenarios such as hypersonic flow fields with unknown perturbations and unmodeled magnetohydrodynamics.
[0005] To automatically extract explicit physical equations from data, symbolic regression has been introduced into mechanism discovery tasks. Traditional genetic programming symbolic regression methods encode mathematical formulas into abstract syntax trees and perform population evolution through random crossover and mutation operators. However, in the huge symbol combination space composed of high-dimensional state variables, this method faces a serious bottleneck in search efficiency. More importantly, the random genetic operations in genetic programming are highly destructive to structure; those high-value substructures with local physical significance that are accidentally combined are often quickly lost in subsequent mutations or crossovers. Existing algorithms lack explicit mechanisms to identify, remember, and reuse these underlying physical building blocks across generations, making it difficult to effectively accumulate knowledge during the search process, easily getting trapped in local optima and converging slowly. Although deep reinforcement learning has been attempted to guide symbolic search, it requires mapping discrete symbol spaces to continuous network parameters and performing massive sampling, making training extremely unstable and heavily reliant on the computing power of large-scale GPU clusters, resulting in high engineering costs.
[0006] Therefore, how to design a mechanism discovery method based on reinforcement learning-inspired genetic programming that can explicitly extract, retain for a long time, and reuse high-value physical substructures across stages, thereby directly reconstructing the true dynamic analytical equations of complex nonlinear systems from multidimensional time-series observation data, is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention
[0007] In view of this, the present invention provides a mechanism discovery method and system based on reinforcement learning-inspired genetic programming, aiming to solve the technical problems of existing deep learning black box models being unable to extract physical mechanisms, physical information neural networks relying on known equations, traditional genetic programming being prone to destroying high-value physical building blocks, and deep reinforcement learning symbolic regression training being unstable and computationally expensive. Thus, it efficiently reconstructs the dynamic evolution laws of complex nonlinear systems with explicit mathematical structures from multidimensional time-series observation data.
[0008] To achieve the above objectives, the present invention adopts the following technical solution:
[0009] In a first aspect, the present invention provides a mechanism discovery method based on reinforcement learning-inspired genetic programming, comprising the following steps: S1. Obtain multidimensional time-series observation data of the target nonlinear system under a preset physical field, and configure the total number of evolutionary stages in genetic programming; S2. At the current evolution stage, construct a mechanism sub-expression pool that includes reusable physical mechanism sub-expressions and novel mechanism sub-expressions; S3. Use the aforementioned mechanism sub-expression pool to perform mechanism-guided initialization on the initial symbolic regression population to generate an initial evolutionary population with physical prior bias. S4. Perform an evolutionary search based on genetic programming on the initial evolutionary population, evaluate and select the fitness based on the dynamic state reconstruction error corresponding to each mechanistic analysis model, and obtain the candidate mechanistic analysis model for the current evolutionary stage. S5. Perform topological decomposition on the elite model in the candidate mechanism analysis model, extract a subset of candidate physical laws and perform multidimensional evaluation, and update the mechanism sub-expression pool for the next evolution stage based on the multidimensional evaluation results. S6. Iterate through steps S2 to S5 until the global fitting error meets the preset convergence threshold to obtain the optimal mechanistic analytical model as the dynamic mechanistic analytical equation of the target nonlinear system, which is used to characterize the mathematical and physical evolution law of the target nonlinear system.
[0010] Preferably, in S1, the acquired multidimensional time-series observation data includes: time-series data of surface heat flux density, pressure, and temperature of the aerodynamic-thermal coupling flow field of a hypersonic vehicle, or time-series data of density, temperature, and magnetic field components of plasma magnetohydrodynamic evolution in a tokamak device, or sampling data of the concentration of each component and the reaction rate over time in a chemical reaction kinetics process; the target nonlinear system is an aerospace aerodynamic system, a fluid dynamics system, or a chemical reaction kinetics system.
[0011] Preferably, in step S2, a reusable physical mechanism sub-expression is constructed. include: Obtain the set of globally optimal elite expressions saved up to the previous evolution stage. ; Each expression τ in the globally optimal elite expression set is decomposed into its subtree structure, and the node length s is collected within a preset interval. The valid subexpressions within constitute the candidate set for deduplication. ; Calculate the candidate set Average fitness of each subexpression s and its frequency of occurrence in the set of globally optimal elite expressions ; Based on preset multi-dimensional ranking rules, from the candidate set The top sub-expression is selected as a reusable physical mechanism sub-expression.
[0012] Preferably, the multi-dimensional sorting rule is expressed as follows:
[0013] According to the node length of the subexpression Average fitness Frequency of occurrence Sort in descending order; and if the current candidate subexpression is a strict subtree of any retained subexpression and its overall ranking is lower than a preset threshold, then discard it and prioritize retaining long-range coupling terms with complete physical meaning.
[0014] Preferably, in step S2, a novel mechanism sub-expression is constructed. include: Maintain a collection of historical records Used to record subexpressions that have been used in all previous evolutionary stages:
[0015] in, The set of sub-expressions generated at the t-th evolution stage; From those that conform to preset syntax constraints and whose tree depth is within a preset range Random sampling is performed in the mechanism sub-expression space within the set, and the historical record set is removed. Based on existing sampling results, novel exploratory mechanistic features were obtained:
[0016] in, and These are the lower and upper bounds of the length of the novel subexpression node, respectively.
[0017] Preferably, in step S3, the mechanism-guided initialization includes: Generate a base population consisting of a completely randomized expression tree. , where N is the population size; For each individual in the basic population, generate a random variable that follows a Bernoulli distribution. To control the preset mechanism replacement probability ; If the random variable Then, a sub-expression is sampled from the sub-expression pool of the mechanism, and used to replace the randomly selected subtree position in the current individual to obtain the replacement individual. ; Check the replacement individual Does it meet the preset syntax validity constraints? Thus, the initial individual is obtained. This constitutes the initial evolutionary population.
[0018] Preferably, in step S4, the dynamic state reconstruction error is calculated based on the root mean square residual:
[0019] in, For a multi-dimensional time-series observation dataset of the target nonlinear system, Let D be the number of samples in dataset D. For the observed variables of system state evolution, To and The corresponding input coupling variables, This indicates that the current evaluation mechanism analysis model is based on a given input. The reconstructed output below.
[0020] Preferably, S5 includes: Obtain the top K elite parsing models with the highest fitness in the entire population generated at the current evolutionary stage, and represent each elite parsing model as an abstract syntax tree; The abstract syntax tree is preorder traversed and decomposed into independent subtree topological fragments, with each subtree topological fragment serving as a subset of candidate physical laws. The candidate physical law subset is evaluated in multiple dimensions based on preset evaluation metrics, including the node length, average fitness, and frequency of occurrence of the sub-expression. The subset of qualified candidate physical laws is updated to the mechanism sub-expression pool for the next evolution stage.
[0021] Preferably, it also includes setting anti-stagnation rules based on search feedback: if the reusable physical mechanism sub-expressions in the mechanism sub-expression pool remain unchanged within a consecutive preset number of evolutionary stages, then the current mechanism sub-expression pool is cleared, only the single benchmark individual with the highest fitness in the globally optimal elite model set is retained, and the population and evolutionary search process are reinitialized.
[0022] Secondly, the present invention provides a mechanism discovery system based on reinforcement learning-inspired genetic programming, comprising: Data acquisition and configuration module: used to acquire multidimensional time-series observation data of the target nonlinear system under a preset physical field, and to configure the total number of evolutionary stages in genetic programming; Mechanism Sub-Expression Pool Building Module: Used to build a mechanism sub-expression pool that includes reusable physical mechanism sub-expressions and novel mechanism sub-expressions at the current evolution stage; Mechanism-guided initialization module: used to perform mechanism-guided initialization on the initial symbolic regression population using the mechanism sub-expression pool, generating an initial evolutionary population with physical prior bias; Genetic programming evolution search module: used to perform genetic programming-based evolution search on the initial evolutionary population, evaluate and select fitness based on the dynamic state reconstruction error corresponding to each mechanistic analysis model, and obtain candidate mechanistic analysis models for the current evolutionary stage; Topology decomposition and pool update module: used to decompose the elite model in the candidate mechanism analysis model, extract a subset of candidate physical laws and perform multidimensional evaluation, and update the mechanism sub-expression pool for the next evolution stage based on the multidimensional evaluation results; Convergence determination and equation output module: used to iteratively execute the steps from constructing the mechanism sub-expression pool to updating the mechanism sub-expression pool until the global fitting error meets the preset convergence threshold, and obtain the optimal mechanism analysis model as the dynamic mechanism analysis equation of the target nonlinear system, which is used to characterize the mathematical and physical evolution law of the target nonlinear system.
[0023] As can be seen from the above technical solution, compared with the prior art, the technical solution of the present invention has the following beneficial effects: 1. This method outputs explicit analytical equations composed of mathematical operators and state variables through genetic programming, avoiding the defect of uninterpretable parameters in deep neural network black box models; the obtained dynamic mechanism analytical equations can clearly characterize the causal relationship and coupling mechanism between multidimensional physical quantities, meet the theoretical verification needs of engineering and scientific fields for physical consistency, and can be directly used for system behavior prediction and control law interpretation.
[0024] 2. By dynamically maintaining pools of reusable physical mechanism sub-expressions and pools of novel physical mechanism sub-expressions, high-fitness contributing physical substructures are explicitly extracted, memorized, and reused across stages at the evolutionary stage level. Simultaneously, guided initialization is used to implant prior physical knowledge into the initial population. This mechanism effectively prevents high-value mechanism building blocks from being destroyed during crossover and mutation operations, reduces the redundant search range of symbolic regression, and accelerates the convergence process of the control equations for complex systems.
[0025] 3. A novel subexpression random sampling mechanism based on historical records is introduced into the mechanism subexpression pool to enforce the ability to explore unknown mathematical structures. Simultaneously, an anti-stagnation rule based on search feedback is implemented, clearing the current pool and resetting the population when a reusable component has not been updated for an extended period. This architecture balances the utilization of known physical knowledge with the exploration of unknown structures, significantly improving the probability of the algorithm escaping local optima and ensuring continuous optimization capabilities during the evolutionary process. Attached Figure Description
[0026] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.
[0027] Figure 1A flowchart illustrating a mechanism discovery method based on reinforcement learning-inspired genetic programming, provided in an embodiment of the present invention; Figure 2 This is a schematic diagram illustrating the symbolic syntax tree topology decomposition and feature reuse process provided in an embodiment of the present invention; Figure 3 This is a framework diagram of a mechanism discovery system based on reinforcement learning-inspired genetic programming, provided for an embodiment of the present invention. Detailed Implementation
[0028] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.
[0029] Example 1; like Figure 1 As shown, this embodiment provides a mechanism discovery method based on reinforcement learning-inspired genetic programming, including the following steps: S1. Obtain multidimensional time-series observation data of the target nonlinear system under a preset physical field, and configure the total number of evolutionary stages in genetic programming; S2. At the current evolution stage, construct a mechanism sub-expression pool that includes reusable physical mechanism sub-expressions and novel mechanism sub-expressions; S3. Use the aforementioned mechanism sub-expression pool to perform mechanism-guided initialization on the initial symbolic regression population to generate an initial evolutionary population with physical prior bias. S4. Perform an evolutionary search based on genetic programming on the initial evolutionary population, evaluate and select the fitness based on the dynamic state reconstruction error corresponding to each mechanistic analysis model, and obtain the candidate mechanistic analysis model for the current evolutionary stage. S5. Perform topological decomposition on the elite model in the candidate mechanism analysis model, extract a subset of candidate physical laws and perform multidimensional evaluation, and update the mechanism sub-expression pool for the next evolution stage based on the multidimensional evaluation results. S6. Iterate through steps S2 to S5 until the global fitting error meets the preset convergence threshold to obtain the optimal mechanistic analytical model as the dynamic mechanistic analytical equation of the target nonlinear system, which is used to characterize the mathematical and physical evolution law of the target nonlinear system.
[0030] By constructing a dynamic mechanism sub-expression pool based on reusable and novel mechanistic sub-expressions, and combining mechanism-guided initialization, genetic programming evolutionary search, and topological decomposition and feedback updates of elite models, this method achieves white-box automatic discovery of the underlying dynamic equations of complex nonlinear systems. This method effectively preserves the high-value physical substructures generated during the evolution process, avoids the problem of easily destroying local feasible solutions in traditional genetic programming, reduces the search redundancy of symbolic regression in high-dimensional space, and improves the convergence efficiency and interpretability of mechanistic analytical models.
[0031] The following provides further explanation of each step in the above method and its related technical features; In this embodiment, S1, multi-dimensional time-series observation data of the target nonlinear system under a preset physical field are obtained, and the total number of evolutionary stages of genetic programming is configured. The acquired multidimensional time-series observation data includes: time-series data of surface heat flux density, pressure, and temperature of the aerodynamic-thermal coupling flow field of a hypersonic vehicle, or time-series data of density, temperature, and magnetic field components of plasma magnetohydrodynamic evolution in a tokamak device, or sampling data of the concentration of each component and the reaction rate over time during chemical reaction kinetics; the target nonlinear system is an aerospace aerodynamic system, a fluid dynamics system, or a chemical reaction kinetics system.
[0032] This embodiment takes hypersonic aircraft in aerospace aerodynamic systems as the specific application object. The following steps S2 to S6 are all elaborated in detail on the discovery of the aerodynamic-thermal coupling flow field mechanism in this scenario. The aim is to automatically extract the explicit analytical equations governing the thermodynamic evolution from the time-series data such as heat flux density, pressure, and temperature collected by sensors on the surface of the aircraft.
[0033] In this embodiment S2, at the current evolution stage, a mechanism sub-expression pool is constructed, including reusable physical mechanism sub-expressions and novel mechanism sub-expressions; Among them, constructing reusable physical mechanism sub-expressions include: Obtain the set of globally optimal elite expressions saved up to the previous evolution stage. ; Each expression τ in the globally optimal elite expression set is decomposed into its subtree structure, and the node length s is collected within a preset interval. The valid subexpressions within constitute the candidate set for deduplication. :
[0034] in, and These represent the lower and upper bounds of the length of the subexpression node, respectively. Indicates from the expression The operation of extracting all subtrees whose node lengths fall within the specified interval; Calculate the candidate set Average fitness of each subexpression s and its frequency of occurrence in the set of globally optimal elite expressions The average fitness is the average fitness of all elite individuals containing this subexpression.
[0035]
[0036] in, For individuals fitness function, express yes subtree, For indicator functions; Based on preset multi-dimensional ranking rules, from the candidate set The top sub-expression is selected as a reusable physical mechanism sub-expression.
[0037] Furthermore, the multi-dimensional ranking rule is expressed as follows:
[0038] According to the node length of the subexpression Average fitness Frequency of occurrence Sort in descending order; and if the current candidate subexpression is a strict subtree of any retained subexpression and its overall ranking is lower than a preset threshold, then discard it and prioritize retaining long-range coupling terms with complete physical meaning.
[0039] Furthermore, construct novel mechanism sub-expressions. include: Maintain a collection of historical records Used to record subexpressions that have been used in all previous evolutionary stages:
[0040] in, The set of sub-expressions generated at the t-th evolution stage; From those that conform to preset syntax constraints and whose tree depth is within a preset range Random sampling is performed in the mechanism sub-expression space within the set, and the historical record set is removed. Based on existing sampling results, novel exploratory mechanistic features were obtained:
[0041] in, and These are the lower and upper bounds of the length of the novel subexpression node, respectively.
[0042] Furthermore, it also includes setting anti-stagnation rules based on search feedback: if the reusable physical mechanism sub-expressions in the mechanism sub-expression pool remain unchanged within a consecutive preset number of evolutionary stages, then the current mechanism sub-expression pool is cleared, only the single benchmark individual with the highest fitness in the globally optimal elite model set is retained, and the population and evolutionary search process are reinitialized.
[0043] For hypersonic vehicles, when constructing the mechanistic sub-expression pool, the system decomposes typical sub-fragments that may constitute the final heat flow equation from the elite model retained in the previous stage, such as the dynamic pressure term. Fourier heat conduction term Radiative heat transfer and the chemical nonequilibrium index term At the same time, randomly sample sub-expressions that have not appeared before (such as...). , (etc.) are added to the novel pool; these sub-expressions are the basic building blocks of the three terms in the final optimal model: convection-diffusion coupling, radiation heat transfer, and chemical catalysis correction, providing clear physical priors for subsequent evolution.
[0044] In this embodiment S3, the initial symbolic regression population is initialized using the mechanistic sub-expression pool to generate an initial evolutionary population with physical prior bias. Mechanism-guided initialization includes: Generate a base population consisting of a completely randomized expression tree. , where N is the population size; For each individual in the basic population, generate a random variable that follows a Bernoulli distribution. To control the preset mechanism replacement probability ; If the random variable Then, a sub-expression is sampled from the sub-expression pool of the mechanism, and used to replace the randomly selected subtree position in the current individual to obtain the replacement individual. ; Check the replacement individual Does it meet the preset syntax validity constraints? Thus, the initial individual is obtained. :
[0045] All initial individuals constitute an initial evolutionary population with physical prior bias.
[0046] In hypersonic vehicle modeling, the system generates 1000 random expression trees as the basic population. Then, with a 30% probability, sub-expressions are extracted from the mechanistic sub-expression pool to replace the random sub-trees in the individuals. In this way, a large number of individuals in the initial population naturally carry the local structure of the final trinomial model. For example, some individuals already contain the prototype of convection-diffusion coupling terms, while others embed radiation terms or exponential catalysis terms. This initialization method enables the population to converge towards the final trinomial analytical equation from the beginning, reducing the proportion of obviously non-physical expressions generated in subsequent evaluations.
[0047] In this embodiment, S4, an evolutionary search based on genetic programming is performed on the initial evolutionary population. Fitness is evaluated and selected based on the dynamic state reconstruction error corresponding to each mechanistic analytical model to obtain candidate mechanistic analytical models for the current evolutionary stage. Specifically, this includes: Calculate the root mean square residual of the mechanistic analysis model for each individual on the observed data, and use its reciprocal as the fitness; select parent individuals based on fitness, and perform crossover and mutation operations on the parent individuals to generate offspring; repeat the above process until the population fitness converges; select the individual with the highest fitness in the current population as the candidate mechanistic analysis model for the current evolutionary stage. The dynamic state reconstruction error is calculated based on the root mean square residual:
[0048] in, For a multi-dimensional time-series observation dataset of the target nonlinear system, Let D be the number of samples in dataset D. For the observed variables of system state evolution, To and The corresponding input coupling variables, This indicates that the current evaluation mechanism analysis model is based on a given input. The reconstructed output below; For hypersonic vehicle heat flux prediction, each candidate model The input is taken from wind tunnel or telemetry data, and the output is the predicted heat flow. The system uses the reciprocal of the root mean square residual as fitness and iteratively optimizes it through tournament selection, subtree crossover, and node mutation. As evolution progresses, the individual with the highest fitness gradually evolves from a chaotic expression to a three-term dominant form: one term containing... One is The other one is The candidate models at this stage are very close to the final optimal mechanistic analysis model.
[0049] In this embodiment, S5, the elite model in the candidate mechanism analysis model is topologically decomposed, a subset of candidate physical laws is extracted and multidimensionally evaluated, and the mechanism sub-expression pool for the next evolution stage is updated based on the multidimensional evaluation results; including: Obtain the top K elite parsing models with the highest fitness in the entire population generated at the current evolutionary stage, and represent each elite parsing model as an abstract syntax tree; The abstract syntax tree is preorder traversed and decomposed into independent subtree topological fragments, with each subtree topological fragment serving as a subset of candidate physical laws. The candidate physical law subset is evaluated in multiple dimensions based on preset evaluation metrics, including the node length, average fitness, and frequency of occurrence of the sub-expression. The subset of qualified candidate physical laws is updated to the mechanism sub-expression pool for the next evolution stage.
[0050] In this step, after obtaining multiple elite models for the current stage, the system converts them one by one into abstract syntax trees and decomposes all subtree fragments through preorder traversal; if an elite model is in the form of... The system will extract and Calculate the node length, average fitness, and frequency of occurrence for each fragment; sort the fragments in descending order according to the rule of node length first, average fitness second, and frequency third, and retain the first 5 fragments as a subset of reusable physics rules; For hypersonic vehicles, segments with long node lengths often characterize macroscopic Reynolds analogies and are therefore given priority. Simple terms that appear frequently but are short, such as ρ or u, may be discarded because they are too basic and lack specificity. Finally, qualified sub-segments are stored in the mechanism sub-expression pool for the next stage, so that subsequent evolution can directly call these verified components to construct more complex equations.
[0051] In this embodiment, steps S6 and S2 to S5 are executed iteratively until the global fitting error meets the preset convergence threshold, thereby obtaining the optimal mechanism analysis model as the dynamic mechanism analysis equation of the target nonlinear system, which is used to characterize the mathematical and physical evolution law of the target nonlinear system. Here, the global fitting error is measured by the root mean square error or the mean absolute percentage error, and the convergence threshold is the preset upper limit of error or the error change rate of multiple consecutive evolution stages is less than the preset percentage threshold. like Figure 2As shown, a closed-loop reuse mechanism between the physical mechanism sub-expression pool and genetic programming search is demonstrated: the mechanism sub-expression pool consists of reusable physical mechanism sub-expressions and novel mechanism sub-expressions, wherein the reusable physical mechanism sub-expressions are derived from the topological decomposition and extraction of elite models, and the novel mechanism sub-expressions are obtained by randomly sampling unvisited structures in the expression space. At each evolutionary stage, the system uses two types of sub-expressions in the pool to perform local subtree replacement on the randomly generated initial population, completing mechanism-guided initialization and generating an initial evolutionary population with physical prior bias. Subsequently, new elite expressions are generated through genetic programming search and stored in the global optimal set. These elite expressions are then decomposed and extracted again to update the reusable physical mechanism sub-expressions in the sub-expression pool, thus forming a closed-loop strategy of extraction—pool update—guided initialization—evolutionary search—re-extraction. Through explicit feature pooling and cross-stage reuse mechanism, it achieves long-term memory and iterative reinforcement of high-value physical components. Furthermore, the system repeats S2 to S5 in stages; after each stage, the global root mean square error of the current best candidate model is calculated on the entire spacecraft observation dataset. The preset convergence condition is: Or the error reduction ratio is less than 1% for three consecutive stages; if not, the evolution continues to the next stage; during the iteration process, the system also monitors the update status of reusable parts in the mechanism sub-expression pool in real time: if no new components are added to the pool for five consecutive stages, it is determined that the search has fallen into a local optimum; at this time, the anti-stagnation rule is triggered: clear all components in the current reusable pool, retain only the single individual with the highest fitness in the global history as the seed, and then regenerate a diversified initial population to escape the current local extreme region; After several stages, when the global fitting error stabilizes below the convergence threshold, the system terminates the iteration and outputs the final optimal mechanistic analytical model; this model is the explicit equation for the automatic discovery of the aerodynamic-thermal coupled flow field of hypersonic vehicles:
[0052] in, The wall heat flux density, , , , , These are density, velocity, temperature, wall temperature, and ambient temperature, respectively. T is the normal temperature gradient of the wall. , , These are explicit constants that are automatically determined through the genetic programming and reinforcement learning-inspired search.
[0053] The final optimal mechanism analysis model has clear technical advantages; each term in the equation corresponds to a real physical process, and engineers can directly view the mathematical form of each term to determine the model's effectiveness under flight conditions; moreover, the model only contains multiplication, exponentiation, and exponential functions, with a single calculation taking microseconds, fully meeting the millisecond-level real-time prediction requirements of the aircraft's onboard computer; in addition, the model does not rely on any external solvers or neural network libraries, and can be written as a few lines of C code and directly burned into the flight control chip, without worrying about software environment compatibility issues; In thermal protection engineering, this model can support a variety of typical applications. During flight, sensors measure parameters such as incoming flow density, velocity, and wall temperature in real time. Substituting these parameters into the model immediately yields the current heat flow on the wall, allowing the control system to adjust the spray flow rate of the sweating coolant in advance. Furthermore, by combining the future flight trajectory provided by the navigation system, the model can quickly scan the peak heat flow along the entire trajectory. Once the temperature approaches the structural temperature limit, an alarm is issued, guiding the aircraft to actively avoid dangerous conditions. In addition, by inputting the heat flow predicted by the model as a boundary condition into a simplified heat conduction model, the temperature distribution and ablation depth inside the heat protection layer can be calculated in real time during flight, helping to determine whether reusable aircraft need to be repaired in advance.
[0054] This embodiment takes the aero-thermal coupling flow field of a hypersonic vehicle as a specific application object. By constructing a dynamic mechanism sub-expression pool, mechanism-guided initialization, genetic programming evolution search, and elite model topology decomposition and feedback update, it realizes the automatic extraction of white-box dynamic analytical equations from time-series observation data such as heat flux density, pressure, and temperature on the vehicle surface. The final output explicit mathematical physics equations do not contain any uninterpretable black-box parameters and can be directly used for online thermal environment prediction and control decisions of the vehicle's thermal protection system, providing reliable mechanistic analysis support for the thermal safety design and adaptive flight of hypersonic vehicles.
[0055] Example 2; like Figure 3 As shown, this embodiment provides a mechanism discovery system based on reinforcement learning-inspired genetic programming, including: Data acquisition and configuration module: used to acquire multidimensional time-series observation data of the target nonlinear system under a preset physical field, and to configure the total number of evolutionary stages in genetic programming; Mechanism Sub-Expression Pool Building Module: Used to build a mechanism sub-expression pool that includes reusable physical mechanism sub-expressions and novel mechanism sub-expressions at the current evolution stage; Mechanism-guided initialization module: used to perform mechanism-guided initialization on the initial symbolic regression population using the mechanism sub-expression pool, generating an initial evolutionary population with physical prior bias; Genetic programming evolution search module: used to perform genetic programming-based evolution search on the initial evolutionary population, evaluate and select fitness based on the dynamic state reconstruction error corresponding to each mechanistic analysis model, and obtain candidate mechanistic analysis models for the current evolutionary stage; Topology decomposition and pool update module: used to decompose the elite model in the candidate mechanism analysis model, extract a subset of candidate physical laws and perform multidimensional evaluation, and update the mechanism sub-expression pool for the next evolution stage based on the multidimensional evaluation results; Convergence determination and equation output module: used to iteratively execute the steps from constructing the mechanism sub-expression pool to updating the mechanism sub-expression pool until the global fitting error meets the preset convergence threshold, and obtain the optimal mechanism analysis model as the dynamic mechanism analysis equation of the target nonlinear system, which is used to characterize the mathematical and physical evolution law of the target nonlinear system.
[0056] The hypersonic vehicle aerothermal mechanism discovery system provided in this embodiment can acquire time-series data such as heat flux, pressure, and temperature on the vehicle surface through a data acquisition and configuration module. Then, the mechanism sub-expression pool construction module, guided initialization module, genetic programming evolution search module, and topology decomposition and pool update module work together to form a closed loop of physical feature extraction and reuse across stages. Finally, the convergence judgment and equation output module obtains explicit analytical equations without black-box parameters. This system can be directly embedded into the vehicle's thermal protection system for real-time heat flux prediction and trajectory optimization, providing a complete automated solution for mechanism modeling in complex thermal environments of hypersonic vehicles.
[0057] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the systems disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the descriptions are relatively simple; relevant parts can be referred to the method section.
[0058] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.
Claims
1. A mechanism discovery method based on reinforcement learning inspired genetic programming, characterized by, Includes the following steps: S1. Obtain multidimensional time-series observation data of the target nonlinear system under a preset physical field, and configure the total number of evolutionary stages in genetic programming; S2. At the current evolution stage, construct a mechanism sub-expression pool that includes reusable physical mechanism sub-expressions and novel mechanism sub-expressions; S3. Use the aforementioned mechanism sub-expression pool to perform mechanism-guided initialization on the initial symbolic regression population to generate an initial evolutionary population with physical prior bias. S4. Perform an evolutionary search based on genetic programming on the initial evolutionary population, evaluate and select the fitness based on the dynamic state reconstruction error corresponding to each mechanistic analysis model, and obtain the candidate mechanistic analysis model for the current evolutionary stage. S5. Perform topological decomposition on the elite model in the candidate mechanism analysis model, extract a subset of candidate physical laws and perform multidimensional evaluation, and update the mechanism sub-expression pool for the next evolution stage based on the multidimensional evaluation results. S6. Iterate through steps S2 to S5 until the global fitting error meets the preset convergence threshold to obtain the optimal mechanistic analytical model as the dynamic mechanistic analytical equation of the target nonlinear system, which is used to characterize the mathematical and physical evolution law of the target nonlinear system. 2.The mechanism discovery method based on reinforcement learning inspired genetic programming according to claim 1, wherein, In S1, the acquired multidimensional time-series observation data includes: time-series data of surface heat flux density, pressure, and temperature of the aerodynamic-thermal coupling flow field of a hypersonic vehicle, or time-series data of density, temperature, and magnetic field components of plasma magnetohydrodynamic evolution in a tokamak device, or sampling data of the concentration of each component and the reaction rate over time during chemical reaction kinetics; the target nonlinear system is an aerospace aerodynamic system, a fluid dynamics system, or a chemical reaction kinetics system.
3. The mechanism discovery method based on reinforcement learning-inspired genetic programming according to claim 1, characterized in that, In the S2, a reusable physical mechanism sub-expression is constructed comprising: obtaining a set of global optimal elite expressions saved up to a previous evolution stage ; decompose each expression τ in the global optimal elite expression set into its sub-tree structure, collect valid sub-expressions with node length s in a preset interval to form a deduplication candidate set ; computing the candidate set average fitness of each sub-expression s and its frequency of occurrence in the global optimal elite expression set ; Based on a preset multi-dimension sorting rule, a top sub-expression is filtered from the candidate set as a reusable physical mechanism sub-expression.
4. The mechanism discovery method based on reinforcement learning inspired genetic programming according to claim 3, characterized in that, The multi-dimensional sorting rule is expressed as follows: in order of the node length of the sub-expressions , average fitness , frequency of occurrence in descending order; and, if the current candidate sub-expression is a strict sub-tree of any of the sub-expressions that have been reserved and its overall rank is lower than a preset threshold, it is rejected, and the long-range coupling term with the complete physical meaning is preferentially reserved.
5. The mechanism discovery method based on reinforcement learning-inspired genetic programming according to claim 1, characterized in that, In the S2, a novel mechanism sub-expression is constructed comprising: maintaining a history set for recording all previously used sub-expressions of the evolution stages wherein, a set of sub-expressions generated for the tth evolution stage; randomly sample from a mechanism sub-expression space that conforms to pre-set syntax constraints and has a tree depth within a pre-set interval and remove the sampled results that already exist in the history record set to obtain exploratory novel mechanism features wherein, and are the lower and upper bounds, respectively, of the novel subexpression node length.
6. The mechanism discovery method based on reinforcement learning inspired genetic programming according to claim 1, characterized in that, In step S3, the mechanism-guided initialization includes: Generating a base population consisting of fully random expression trees where N is the population size; generating a random variable subject to a Bernoulli distribution for each individual in the base population to replace the probability of the preset mechanism in control ; If the random variable Then, a sub-expression is sampled from the sub-expression pool of the mechanism, and used to replace the randomly selected subtree position in the current individual to obtain the replacement individual. ; Check the replacement individual Does it meet the preset syntax validity constraints? Thus, the initial individual is obtained. This constitutes the initial evolutionary population.
7. The mechanism discovery method based on reinforcement learning-inspired genetic programming according to claim 1, characterized in that, In step S4, the dynamic state reconstruction error is calculated based on the root mean square residual: in, For a multi-dimensional time-series observation dataset of the target nonlinear system, Let D be the number of samples in dataset D. For the observed variables of system state evolution, To and The corresponding input coupling variables, This indicates that the current evaluation mechanism analysis model is based on a given input. The reconstructed output below.
8. The mechanism discovery method based on reinforcement learning-inspired genetic programming according to claim 1, characterized in that, S5 includes: Obtain the top K elite parsing models with the highest fitness in the entire population generated at the current evolutionary stage, and represent each elite parsing model as an abstract syntax tree; The abstract syntax tree is preorder traversed and decomposed into independent subtree topological fragments, with each subtree topological fragment serving as a subset of candidate physical laws. The candidate physical law subset is evaluated in multiple dimensions based on preset evaluation metrics, including the node length, average fitness, and frequency of occurrence of the sub-expression. The subset of qualified candidate physical laws is updated to the mechanism sub-expression pool for the next evolution stage.
9. The mechanism discovery method based on reinforcement learning-inspired genetic programming according to claim 1, characterized in that, It also includes setting anti-stagnation rules based on search feedback: if the reusable physical mechanism sub-expressions in the mechanism sub-expression pool remain unchanged within a preset number of consecutive evolutionary stages, then the current mechanism sub-expression pool is cleared, only the single benchmark individual with the highest fitness in the globally optimal elite model set is retained, and the population and evolutionary search process are reinitialized.
10. A mechanism discovery system based on reinforcement learning-inspired genetic programming, characterized in that, include: Data acquisition and configuration module: used to acquire multidimensional time-series observation data of the target nonlinear system under a preset physical field, and to configure the total number of evolutionary stages in genetic programming; Mechanism Sub-Expression Pool Building Module: Used to build a mechanism sub-expression pool that includes reusable physical mechanism sub-expressions and novel mechanism sub-expressions at the current evolution stage; Mechanism-guided initialization module: used to perform mechanism-guided initialization of the initial symbolic regression population using the mechanism sub-expression pool, generating an initial evolutionary population with physical prior bias; Genetic programming evolution search module: used to perform genetic programming-based evolution search on the initial evolutionary population, evaluate and select fitness based on the dynamic state reconstruction error corresponding to each mechanistic analysis model, and obtain candidate mechanistic analysis models for the current evolutionary stage; Topology decomposition and pool update module: used to decompose the elite model in the candidate mechanism analysis model, extract a subset of candidate physical laws and perform multidimensional evaluation, and update the mechanism sub-expression pool for the next evolution stage based on the multidimensional evaluation results; Convergence determination and equation output module: used to iteratively execute the steps from constructing the mechanism sub-expression pool to updating the mechanism sub-expression pool until the global fitting error meets the preset convergence threshold, and obtain the optimal mechanism analysis model as the dynamic mechanism analysis equation of the target nonlinear system, which is used to characterize the mathematical and physical evolution law of the target nonlinear system.