A physical field reconstruction method and apparatus

Abstract

The present application relates to the technical field of physical field reconstruction, and provides a physical field reconstruction method and device, comprising: obtaining surface mesh data of a target object; based on the surface mesh data, performing a batch optimization sampling strategy to determine a sampling point position; based on the sampling point position, constructing a proxy model, when a model parameter matrix is numerically unstable, sequentially degrading a kernel function in a preset order until numerically stable; based on the proxy model, reconstructing a full-field stress field and quantifying a prediction confidence. The present application improves the reconstruction accuracy under sparse sampling conditions through the batch optimization sampling strategy, enhances the robustness of the proxy model using a multi-level kernel function degradation mechanism, and provides a strict statistical guarantee for the prediction result by quantifying the prediction confidence.

Description

Technical Field

[0001] This invention relates to the field of physical field reconstruction technology, and specifically to a physical field reconstruction method and apparatus. Background Technology

[0002] In the fields of biomedical engineering and industrial testing, stress field measurement of thin-layer soft tissues (such as periodontal ligaments and other biological soft tissues) is of great significance for medical evaluation and engineering applications. However, due to the structural characteristics of thin-layer soft tissues, directly measuring their complete three-dimensional stress field is extremely difficult. On the one hand, due to anatomical constraints, the number of sensors that can be implanted on the body surface is very limited, typically allowing only a very small number of measurement points to be installed, resulting in stress data that covers only a very small proportion of the total volume. On the other hand, the stress distribution inside thin-layer structures is complex and non-uniform, and the stress tensor itself has multiple components, making the constraint of the high-dimensional output space extremely difficult.

[0003] In existing technologies, finite element method (FEM) simulations or surrogate models are commonly used to reconstruct stress fields. While traditional FEM simulations have a solid physical foundation, they are computationally expensive and struggle to directly integrate sparse in vivo sensor data. Surrogate models, such as Gaussian process regression or neural networks, while computationally efficient, often suffer from low prediction accuracy and poor generalization ability when dealing with very few sampling points and high-dimensional tensor outputs. In particular, standard Gaussian process regression is ill-suited for non-stationary stress fields with large gradient differences between stress concentration and flat regions. Furthermore, existing data-driven surrogate models often lack explicit encoding of physical constraints, leading to potential non-physical oscillations in the prediction results. More importantly, current technologies lack theoretical means to quantify the uncertainty of prediction results, failing to provide confidence assessments with rigorous statistical guarantees, and thus failing to meet the reliability requirements of clinical or engineering decision-making. Summary of the Invention

[0004] To address the technical problems of low accuracy, poor model robustness, and lack of theoretical confidence guarantee in existing technologies for physical field reconstruction under conditions with very few sensing points, this application proposes a physical field reconstruction method and apparatus that achieves high-precision, high-robustness, and theoretically confident physical field reconstruction under sparse sampling conditions.

[0005] To achieve the above objectives, the present invention adopts the following technical solution: A physical field reconstruction method includes: acquiring surface mesh data of a target object; executing a batch optimization sampling strategy based on the surface mesh data to determine the sampling point locations; constructing a surrogate model based on the sampling point locations, and when the model parameter matrix exhibits numerical instability, sequentially downgrading the kernel function in a preset order until the values ​​stabilize; and reconstructing the full-field stress field and quantifying the prediction confidence based on the surrogate model.

[0006] This scheme employs a batch optimization strategy in the sampling phase, which can acquire more information under conditions with very few sensor points compared to the traditional point-by-point greedy method. In the model building phase, a kernel function degradation mechanism is introduced, which automatically adjusts the kernel function when the extreme distribution of sampling points leads to numerical instability, ensuring that the model can work normally under various conditions. At the same time, by quantifying the prediction confidence, it provides a statistical theoretical guarantee for the reconstruction results.

[0007] Preferably, the batch optimization sampling strategy includes: determining a pre-score for candidate sampling points based on the geometric and physical features of the surface mesh data; and performing batch joint optimization based on the pre-score to determine the location of the sampling points.

[0008] Preferably, the geometric features include curvature features and geodesic distance features; the physical features include preload stress features and foundation variance features; and the batch joint optimization uses the posterior variance reduction as the acquisition function.

[0009] Preferably, the surrogate model employs a non-stationary kernel function and incorporates physical prior means.

[0010] Preferably, the non-stationary kernel function is a Gibbs kernel function, and its length scale is adaptively determined by a neural network; the step of sequentially reducing the kernel function according to a preset order includes: when the covariance matrix becomes ill-conditioned, the non-stationary kernel function is sequentially reduced to a stationary kernel function until the covariance matrix is ​​positive definite.

[0011] Preferably, the quantitative prediction confidence level includes: determining the confidence interval of the prediction result based on the upper bound of the generalization error.

[0012] Preferably, the upper bound of the generalization error is determined by: calculating the upper bound of the generalization error based on the KL divergence between the posterior and prior distributions, the number of training samples, and the confidence parameter; the confidence interval is constructed based on the upper bound of the generalization error and the predicted mean.

[0013] Preferably, the acquisition of the surface mesh data of the target object includes: loading a three-dimensional mesh file and performing region filtering based on the coordinate range to obtain candidate sampling regions.

[0014] Preferably, the construction of the surrogate model further includes: obtaining stress tensor data at the sampling point locations; wherein the stress tensor data is obtained through a three-layer adaptive physical simulation, the three-layer adaptive physical simulation including exact solution, intermediate-level solution and fast fallback solution.

[0015] Furthermore, the present invention also provides a physical field reconstruction device, comprising: a data acquisition module configured to acquire surface mesh data of a target object; a sampling module configured to execute a batch optimization sampling strategy based on the surface mesh data to determine the sampling point locations; a model construction module configured to construct a surrogate model based on the sampling point locations, wherein when the model parameter matrix exhibits numerical instability, the kernel function is sequentially downgraded in a preset order until the values ​​stabilize; and a prediction module configured to reconstruct the full-field stress field and quantify the prediction confidence based on the surrogate model.

[0016] Beneficial effects: The technical solution provided by this invention has the following beneficial effects: 1. By implementing a batch optimization sampling strategy, combining geometric and physical features for pre-scoring, and employing batch joint optimization to determine sampling point locations, this invention avoids information redundancy caused by traditional point-by-point greedy sampling. It maximizes information gain with minimal sampling points, thereby improving the accuracy of full-field stress field reconstruction. In practical applications such as thin-layer soft tissue stress field measurement, this invention can accurately reconstruct a complete three-dimensional stress field containing thousands of mesh vertices with only a small number of surface sensors, significantly alleviating the problem of insufficient measurement points due to anatomical location constraints.

[0017] 2. By constructing a surrogate model with a multi-level kernel function degradation mechanism, when the model parameter matrix becomes numerically unstable, it can automatically reduce the kernel function according to a preset order until the values ​​stabilize. This effectively solves the ill-conditioned covariance matrix problem caused by extreme sampling point distribution, ensuring the system's engineering robustness. Simultaneously, by employing a non-stationary kernel function and incorporating physical prior means, the surrogate model can adapt to the non-stationary characteristics of large gradient differences between stress concentration and smooth regions, ensuring the physical rationality of the prediction results. This feature is particularly suitable for scenarios with complex and non-uniform stress distribution within thin soft tissues, avoiding the problems of low prediction accuracy and poor generalization ability that traditional surrogate models encounter when handling high-dimensional tensor outputs.

[0018] 3. By quantifying the prediction confidence level based on the upper bound of the generalization error, a rigorous statistical confidence guarantee is provided for the prediction results, overcoming the lack of theoretical basis in traditional uncertainty quantification methods. In clinical medical evaluation or engineering testing applications, this theoretical guarantee allows doctors or engineers to clearly understand the reliability of the prediction results, providing a reliable reference for subsequent clinical or engineering decisions. Attached Figure Description

[0019] Figure 1 This is a flowchart of the physical field reconstruction method according to an embodiment of the present invention. Detailed Implementation

[0020] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of the embodiments of this invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0021] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used herein in the description of the invention is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention.

[0022] Example 1: As Figure 1 As shown, this embodiment provides a physical field reconstruction method. This method addresses the difficulty of full-field stress reconstruction in complex structures such as thin-layer soft tissues under sparse sensing conditions, and constructs a complete process from data acquisition to confidence assessment. The method includes the following steps: Step S100: Obtain the surface mesh data of the target object.

[0023] Specifically, the target object can be biological soft tissue (such as the periodontal ligament) or an engineered component with a thin-layer structure. Surface mesh data is typically obtained through 3D scanning or medical image reconstruction, containing the coordinate information of mesh vertices and topological connectivity. This data serves as the basic geometric carrier for subsequent sampling and modeling. In this embodiment, the acquired data needs to be preprocessed, such as removing redundant vertices or repairing inconsistencies in normals, to ensure the integrity of the geometric model, thereby providing accurate geometric constraints for subsequent sampling on the manifold surface.

[0024] Step S200: Based on the surface mesh data, execute a batch optimization sampling strategy to determine the sampling point locations.

[0025] This step aims to address the problem of low information acquisition efficiency under sparse sampling conditions. Traditional point-by-point greedy sampling easily leads to sampling point clustering, resulting in information redundancy. The batch optimization sampling strategy adopted in this embodiment treats the sampling process as a global optimization problem, jointly evaluating the information gain of a batch of candidate points rather than evaluating the value of a single point in isolation. Specifically, this strategy comprehensively considers the geometric features (such as curvature distribution) and physical features (such as pre-estimated stress distribution) of the target object, and within a limited sampling budget (e.g., only 8 sensors can be deployed), seeks the combination of locations that maximizes the reduction of overall prediction uncertainty. Through this batch joint evaluation, excessive concentration of sampling points in local areas is effectively avoided, ensuring the uniformity and representativeness of the sampling layout in both geometric and physical feature spaces, thereby providing high-quality sparse data support for subsequent model training.

[0026] Step S300: Based on the sampling point location, construct a proxy model. When the model parameter matrix shows unstable values, downgrade the kernel function in a preset order until the values ​​stabilize.

[0027] After obtaining the physical field data (such as stress tensor) from the sampling points, a surrogate model needs to be constructed to establish the mapping relationship from spatial coordinates to the physical field response. This embodiment uses Gaussian process regression as the basic framework of the surrogate model. However, under sparse sampling and extreme data distribution, the covariance matrix involved in the model training process is prone to ill-conditioned problems (such as non-positive definiteness), leading to model failure. To solve this problem, this embodiment introduces a kernel function degradation mechanism. Specifically, the system presets a degradation order from complex kernel functions to simple kernel functions. During model construction, the numerical stability of the covariance matrix is ​​monitored in real time. Once numerical instability is detected (such as excessive matrix condition number or Cholesky decomposition failure), the system automatically triggers the degradation strategy, replacing the currently used kernel function with a simpler structure and stronger numerical stability until the covariance matrix returns to positive definiteness. This mechanism ensures the engineering robustness of the surrogate model under extreme data conditions and avoids system crashes caused by numerical calculation problems.

[0028] Step S400: Based on the surrogate model, reconstruct the full-field stress field and quantify the prediction confidence.

[0029] Using a trained surrogate model, predictions can be made for any unsampled point on the surface mesh of a target object, thereby reconstructing the complete full-field stress field. Unlike traditional surrogate models that only output the predicted mean, the model in this embodiment can also output a measure of the prediction's uncertainty. Furthermore, this embodiment uses generalization error theory to quantify the confidence level of the prediction results. Specifically, by calculating the upper bound of the generalization error, a theoretically guaranteed confidence interval is constructed. This makes the prediction result not only a numerical value but also includes a statistical indicator of "the degree to which this value is reliable." This is of great significance for medical diagnosis or engineering decision-making; for example, doctors can judge the reliability of the prediction results based on the confidence level, thereby formulating more reasonable treatment plans.

[0030] Through the above steps, this embodiment achieves a highly robust and accurate physical field reconstruction method. The batch optimization sampling strategy ensures the information quality of sparse data, the kernel function degradation mechanism ensures the stability of the model construction process, and the confidence quantification based on generalization error provides theoretical credibility support for the prediction results. These three elements work together to solve the core technical challenges faced in physical field reconstruction under sparse data conditions.

[0031] Example 2: Based on Example 1, this example provides a detailed explanation of the specific implementation process of the batch optimization sampling strategy described in step S200.

[0032] Step S201: Based on the geometric and physical characteristics of the surface mesh data, determine the pre-score of the candidate sampling points.

[0033] To maximize information gain within a limited sampling budget, this embodiment employs a four-dimensional feature fusion pre-scoring mechanism to initially screen and evaluate candidate sampling points from both geometric and physical dimensions. The geometric features include curvature features and geodesic distance features; the physical features include preload stress features and foundation variance features.

[0034] For curvature features, the system calculates the average curvature of the grid vertices. In the stress field of biological soft tissues (such as the periodontal ligament), high curvature regions (such as the cervical margin of the tooth) are usually accompanied by high stress gradients and are high-incidence areas of stress concentration. Therefore, assigning a higher pre-score to high curvature regions can guide the sampling strategy to prioritize these geometrically complex and mechanically critical areas.

[0035] For the geodesic distance feature, the system calculates the surface geodesic distance between candidate points and the selected point set. Unlike Euclidean distance, geodesic distance accurately reflects the true distance of points on the curved manifold. By introducing a geodesic distance penalty term, excessive clustering of sampling points in local areas can be effectively suppressed, promoting a more uniform coverage of the sampling layout on the manifold surface and avoiding "blind spots".

[0036] For preload stress characteristics, the system uses a simplified physical simulation model (such as a linear elastic model) to quickly calculate the preload stress amplitude at candidate points. This characteristic reflects the physical response intensity of the point under initial loading conditions. Points with larger preload stresses often correspond to potential mechanically sensitive areas, and prioritizing sampling of these points helps to capture the main trend of stress distribution across the entire field.

[0037] For the basic variance feature, the system uses a prior surrogate model (such as an untrained Gaussian process model) to evaluate the initial prediction uncertainty of candidate points. Points with larger basic variances mean that the model has lower confidence in predicting at that location, and prioritizing sampling these points can effectively reduce the global uncertainty of the model.

[0038] By comprehensively scoring the above four-dimensional features, the system can generate a candidate point ranking that takes into account both geometric coverage and physical sensitivity, providing a high-quality search space for subsequent batch optimization.

[0039] Step S202: Based on the pre-scoring, perform batch joint optimization to determine the sampling point location.

[0040] Specifically, this embodiment abandons the traditional point-by-point greedy sampling strategy and instead adopts a batch joint optimization strategy. The traditional point-by-point greedy strategy selects only the currently optimal single point at each step, which is prone to getting trapped in local optima, causing multiple sampling points to cluster in the same high-value area, resulting in information redundancy. For example, if a certain area has extremely high curvature, the greedy strategy may continuously select multiple points in that area, while ignoring other areas that, although suboptimal, provide complementary information.

[0041] To address the aforementioned issues, the batch joint optimization employs the posterior variance reduction as the acquisition function.

[0042] The mathematical expression of this sampling function aims to find a set of sampling points such that adding these points maximizes the decrease in the posterior variance integral of the model over the entire domain. Its mathematical form can be expressed as:

[0043] in, The qNegIntegratedPosteriorVariance acquisition function is used to evaluate the overall information gain of a batch of sampling points; This represents a set of sampling points to be determined. Represents the entire surface mesh of the target object; Indicates position The posterior variance at the location; It is a spatial coordinate vector; Indicates an existing dataset; integral symbol This indicates integration over the entire domain. Essentially, the formula evaluates how much the uncertainty in predicting the entire physical field would be reduced if sampling were performed at these locations.

[0044] By maximizing this acquisition function, the system no longer evaluates the value of a single point in isolation, but rather jointly assesses the overall contribution of a batch of points. This makes the selected sampling point combinations inherently mutually exclusive and complementary in space: when a point is selected into the batch, the marginal information gain of its neighboring points automatically decreases, thereby guiding the system to explore other uncovered areas. This mechanism ensures that, under the constraint of a very small number of sampling points (e.g., 8 points), the sampling layout can maximally reproduce the distribution characteristics of the entire stress field, significantly improving the information utilization efficiency of sparse sampling.

[0045] Example 3: This example, based on Example 1, provides a detailed explanation of the specific implementation process of constructing the surrogate model in step S300. Addressing the "non-stationary" characteristics of the stress field in thin-layer soft tissues (i.e., a large gradient difference between stress concentration and smooth areas) and the numerical computational instability that may result from sparse sampling, this example provides a surrogate model construction method with adaptive degradation capabilities and incorporating physical priors.

[0046] Step S301: Construct the surrogate model infrastructure that uses a non-stationary kernel function and incorporates physical prior means.

[0047] Traditional surrogate models (such as standard Gaussian process regression) typically employ stationary kernel functions, assuming that the data has the same correlation length scale throughout the space. However, in the stress fields of biological soft tissues such as the periodontal ligament, there is severe stress concentration in high-curvature regions such as the cervical margin, while other regions are relatively flat. Forcing the use of a single length scale can lead to underfitting in stress-concentrated areas and overfitting in flat areas. Therefore, this embodiment employs a non-stationary kernel function, allowing the correlation length scale to vary with spatial location. Furthermore, to avoid purely data-driven models predicting results that violate fundamental laws of elasticity (such as stress tensor asymmetry), this embodiment incorporates low-precision physical simulation results as prior knowledge into the model, constructing an explicit physical prior mean function.

[0048] Step S302: Determine the specific form of the non-stationary kernel function as the Gibbs kernel function, and adaptively determine its length scale through a neural network.

[0049] This embodiment uses the Gibbs kernel function as a specific implementation of the non-stationary kernel function. This kernel function introduces a length scaling function l(x) that varies with spatial location, and its mathematical expression is as follows:

[0050] in, It is a Gibbs non-stationary kernel function; and It is a spatial coordinate vector; The amplitude of the kernel function is controlled by the signal variance. It is a length scale function, representing spatial location. The rate of correlation decay at a given location; This represents the Euclidean distance between two spatial coordinate vectors. This represents an exponential function. (To achieve length scaling) To achieve adaptive determination, this embodiment introduces a lightweight neural network. The input of this neural network is spatial coordinates. The output is the length scale at that location. By optimizing the logarithmic marginal likelihood function, the neural network can learn to output a smaller length scale in regions of high stress gradients to capture drastic changes, and a larger length scale in regions of gentle stress to smooth out noise. This mechanism enables the surrogate model to adaptively adjust its local fitting ability, thereby accurately describing complex non-stationary stress fields.

[0051] Step S303: Perform the physical prior means fusion operation.

[0052] Specifically, this embodiment abandons the traditional approach of assuming the mean function is zero or constant in Gaussian process regression, and instead uses the low-precision stress field calculated based on a simplified physical model (such as linear elastic finite element simulation) as the prior mean function. The prediction formula for the surrogate model is adjusted as follows:

[0053] in, For position The posterior mean function at point; For position The prior mean function at the location is provided by low-precision physical simulation; This is the kernel vector between the test point and the training point; This represents the transpose of the kernel vector; The kernel matrix between training points; To observe the noise variance; It is the identity matrix; This represents the inverse operation of a matrix; These are the actual observed values ​​at the sampling points; Let be the prior mean vector. This formula shows that the surrogate model is actually learning the residuals between the "true observations" and the "low-precision physical predictions". Since low-precision physical simulations naturally satisfy the equilibrium equations and boundary conditions, this fusion method ensures that the final prediction results conform to physical laws in the overall trend, effectively eliminating non-physical oscillations that may occur in pure data-driven models in sparse data regions.

[0054] Step S304: When the model parameter matrix becomes numerically unstable, the kernel function is downgraded sequentially according to a preset order until the values ​​stabilize.

[0055] Under sparse sampling conditions, if the sampling points are distributed in extreme ways (e.g., multiple points are too close together), the covariance matrix K will become ill-conditioned, meaning the condition number is too large or non-positive definite, leading to Cholesky decomposition failure and the model being unsolvable. To address this engineering challenge, this embodiment designs a five-level kernel function reduction strategy, in the following order: Level 1: Gibbs non-stationary kernel function. This is the preferred kernel function, offering the highest analytical accuracy and adapting to non-stationary characteristics.

[0056] Level 2: Matérn 3 / 2 kernel function. When a Level 1 kernel function causes ill-conditioned matrices, the system automatically downgrades to a Matérn 3 / 2 kernel. This kernel function allows for discontinuous first derivatives of functions and has better numerical stability than the Gibbs kernel.

[0057] Level 3: Matérn 5 / 2 kernel function. If Level 2 is still unstable, it is further downgraded to the Matérn 5 / 2 kernel. This kernel function allows for discontinuities in the second derivative, resulting in stronger smoothness and further improved numerical stability.

[0058] Level 4: RBF (Radial Basis) kernel function. If the aforementioned kernel functions fail, the RBF kernel is used instead. The RBF kernel is infinitely differentiable and is an extremely smooth kernel function, very stable in numerical computation, but it sacrifices the ability to capture local high-frequency changes.

[0059] Level 5: Degenerate primitive kernel with slight white noise perturbation. As a last resort, a slight white noise term is introduced into the RBF kernel, artificially increasing the diagonal elements and forcing the covariance matrix to be positive definite.

[0060] During model building, the system monitors the numerical state of the covariance matrix in real time. If numerical instability is detected (e.g., a NotPositiveDefiniteError exception is thrown by Cholesky decomposition), the system automatically triggers a degradation mechanism, switching kernel functions and recalculating in the aforementioned order until the covariance matrix returns to positive definiteness. This mechanism ensures that the surrogate model can complete training under any sampling layout, greatly improving the system's engineering robustness.

[0061] Example 4: This example, based on Example 1, provides a detailed explanation of the specific implementation process of quantifying the confidence level in step S400. Addressing the problem that traditional uncertainty quantification methods (such as standard deviation) lack statistical theoretical guarantees and are difficult to support key decisions, this example provides a confidence quantification method based on the upper bound of generalization error.

[0062] Step S401: Determine the confidence interval of the prediction result based on the upper bound of the generalization error.

[0063] Traditional surrogate models (such as Gaussian process regression) typically output a predicted mean and a posterior standard deviation. The posterior standard deviation only reflects the uncertainty of the model's internal fit to the data and is not directly equivalent to the model's true generalization error on unseen data. In medical diagnosis or engineering decision-making, the standard deviation alone cannot answer the crucial question of "how much error will the predicted result have with a 95% probability?" Therefore, this embodiment introduces the PAC-Bayes theoretical framework to derive an upper bound on the generalization error applicable to unbounded tensor regression tasks in continuous spaces. This upper bound provides a theoretical limit to the prediction error in probabilistic form, thus constructing a confidence interval with rigorous statistical guarantees.

[0064] Step S402: Calculate the upper bound of the generalization error based on the KL divergence between the posterior and prior distributions, the number of training samples, and the confidence parameter.

[0065] In this embodiment, considering the specific characteristics of stress tensor regression in physical field reconstruction tasks, the following generalization error upper bound formula is constructed:

[0066] in, The stress tensor is represented by the first... The true generalization risk of each component is the expected error of the model on future unknown data. This represents empirical risk, which is the average error of the model on the current training set. These are constant coefficients related to the properties of the loss function; Describe the posterior distribution With prior distribution The KL divergence (Kullback-Leibler Divergence) between the parameters is used to measure the degree of deviation of the model parameters from prior knowledge. Represent the natural logarithm function; This indicates the number of training samples, i.e., the number of sampling points; This represents the confidence level parameter, typically set to 0.05, which signifies a 95% confidence level. This is a constant term related to the properties of the loss function. The formula shows that the true generalization risk consists of three parts: empirical risk, complexity penalty term, and constant term.

[0067] In this formula, KL divergence The calculation of KL divergence is crucial. For the Gaussian process model, both its posterior and prior distributions are normal distributions, and the formula for calculating KL divergence can be simplified to:

[0068] in, Indicates KL divergence; and Representing the stress tensor respectively The posterior and prior distributions of each component; and These are the posterior mean and variance output by the surrogate model, respectively. For prior variance; This represents the natural logarithm function. The formula indicates that the more certain the model's predictions for a given region (…), the better. The smaller the value, or the further the predicted value deviates from the prior mean. The larger the KL divergence, the higher the "risk" of the model in that region, and the greater the corresponding upper bound of the generalization error. This mechanism is intuitive: in sparse data regions, although the model may give a definite prediction, its theoretical risk upper bound will automatically increase due to the large deviation from the prior, thus giving a more conservative confidence interval.

[0069] Step S403: Construct a confidence interval based on the upper bound of the generalization error and the predicted mean.

[0070] Specifically, after calculating the upper bound of the generalization error, this embodiment constructs a confidence interval in the following form:

[0071] in, Indicates the confidence interval; The stress tensor predicted by the surrogate model Each component is at position The mean at; The confidence radius, calculated based on the upper bound of the generalization error, represents the range of uncertainty in the prediction. Unlike traditional confidence intervals based on standard deviation, this interval has a clear probabilistic meaning: within at least... With a probability of 95% (e.g., 95%), the actual stress value will fall within this range. This "theoretical guarantee" is the core advantage of this embodiment. It makes the prediction result no longer just a numerical estimate, but a decision-making basis with a reliable risk boundary. For example, in periodontal ligament stress field reconstruction, dentists can judge the reliability of the predicted stress based on this confidence interval. If the confidence interval is too wide, it suggests that more sampling points need to be added to reduce uncertainty.

[0072] Furthermore, the technical solution of this embodiment also has application scenarios for defensive expansion. Although the above formula is based on Gaussian process regression, its core idea—using KL divergence and sample size to constrain the upper bound of generalization error—is also applicable to other types of surrogate models, such as Bayesian neural networks or variational autoencoders. As long as the model can output the parameters of the posterior and prior distributions, the confidence quantification method provided in this embodiment can be applied. This provides theoretical support for the flexible deployment of the technical solution under different computing power platforms and different accuracy requirements.

[0073] Example 5: Based on Example 1, this example provides a detailed explanation of the specific preprocessing steps for obtaining the surface mesh data of the target object in step S100, and the specific simulation process for obtaining stress tensor data before constructing the surrogate model in step S300. The design of these two steps further enhances the system's automation level and engineering robustness.

[0074] Step S101: Load the 3D mesh file and perform region filtering based on the coordinate range to obtain candidate sampling regions.

[0075] Specifically, the 3D mesh file of the target object is usually derived from medical image reconstruction or 3D scanning, and common formats include STL and OBJ. After the system loads the file, it reads the coordinate information of the mesh vertices. Because the geometry of the target object (such as the periodontal ligament) is complex, and not all areas are suitable for sensor placement or have mechanical research value, region screening is necessary. This embodiment uses a coordinate range-based screening strategy, specifically achieved by setting a Z-axis coordinate threshold. For example, the screening threshold is set as follows:

[0076] in, The filtering threshold for the Z-axis coordinate; and These are the minimum and maximum values ​​of the grid vertices along the Z-axis, respectively. This is a scaling factor, representing the proportion of the filtered area to the total height, typically around 0.5. The system will use a Z-coordinate greater than... The vertices are selected as candidate sampling regions. This operation effectively eliminates regions that contribute little to the mechanical analysis or are anatomically limited (such as the very small area at the root tip or the non-stressed area at the crown), thus focusing the sampling search space on active regions with complex stress distributions. This not only reduces the computational load of subsequent sampling optimization algorithms but also improves the effectiveness and relevance of the sampling points.

[0077] Step S102: Obtain the stress tensor data at the sampling point location.

[0078] After determining the sampling point locations, it is necessary to obtain the true values ​​of the physical fields (i.e., stress tensors) at these points as training labels for the surrogate model. Since complete measured data is often lacking in actual engineering or medical settings, or the cost of such measurements is extremely high, this embodiment uses physical simulation to generate this data. Considering the differences in computing power across different platforms and the potential numerical non-convergence issues during simulation, this embodiment designs a three-layer adaptive physical simulation mechanism to ensure that effective stress data can be output under various extreme conditions.

[0079] The three-layer adaptive physics simulation includes exact solution, intermediate-level solution, and fast fallback solution.

[0080] The first layer is the exact solution. This is the preferred simulation mode, employing a high-fidelity nonlinear finite element solver (such as a solver based on the FEniCSx framework). This mode introduces complex constitutive models (such as a combination of the Ogden hyperelastic model and the Prony viscoelastic model), capable of accurately describing the nonlinear mechanical behavior of biological soft tissues. However, the exact solution has the highest computational cost and depends on specific underlying computational libraries. When the system detects that the underlying dependency libraries are complete and the computing power is sufficient, this layer of solution is executed first. If non-convergence occurs during the solution process or the runtime environment lacks relevant libraries, the system will automatically trigger a degradation mechanism.

[0081] The second layer is the intermediate-level solver. When the exact solution fails, the system seamlessly switches to the intermediate-level solver. This mode employs a customized nonlinear finite element framework, simplifying some complex constitutive descriptions (e.g., using the Mooney-Rivlin model, Ogden hyperelastic constitutive model, etc.). While maintaining a certain level of physical accuracy, it reduces the requirements for the computational environment and improves the stability of the solution. This layer serves as a transition between exact and fast solutions, balancing accuracy and robustness.

[0082] The third layer is a fast fallback solution. This is the final guarantee mechanism. When the above two layers fail, the system will construct a simplified local linear elastic model (e.g., an 8x8x8 hexahedral mesh) and use linear elastic constitutive relations for fast solution. Although this mode has the lowest physical accuracy, its computational cost is extremely low and its numerical stability is extremely high. It can guarantee that a set of stress tensor data that conforms to the basic laws of elasticity will be output under any circumstances, thus avoiding the interruption of the entire reconstruction process due to simulation failure.

[0083] Through the aforementioned three-layer adaptive mechanism, this embodiment constructs a data generation module with "fault tolerance." It can adaptively select the most suitable simulation strategy based on the current hardware environment and numerical calculation status, ensuring that even under extreme conditions such as limited computing power or complex models, the system can still provide continuous and effective training data for the surrogate model, greatly improving the engineering implementation capability of the physics field reconstruction method.

[0084] Example 6: This example provides a physics field reconstruction device for executing the physics field reconstruction method described in the above examples. This device can be integrated into medical diagnostic equipment, industrial testing terminals, or high-performance computing workstations, or it can be deployed as a standalone software system on a cloud server. The device includes: a data acquisition module, a sampling module, a model building module, and a prediction module.

[0085] The data acquisition module is configured to acquire the surface mesh data of the target object.

[0086] Specifically, the data acquisition module is the input interface of the device. Its hardware form can be a network interface card, a USB interface, or a wireless communication module (such as a WiFi module or Bluetooth module), used to receive 3D mesh files (such as STL files) from external storage media (such as USB flash drives or hard drives) or medical imaging equipment (such as dental scanners or CT scanners). At the program level, this module includes a file parsing unit capable of reading mesh vertex coordinates and topological connectivity, and performing the region filtering operation described in Example 5. The data acquisition module encapsulates the processed mesh data into a standard data structure and transmits it to the sampling module.

[0087] The sampling module is configured to execute a batch optimization sampling strategy based on the surface mesh data to determine the sampling point location.

[0088] The sampling module is one of the core computing units of the device. At the hardware level, its functions are implemented by a processor (such as a CPU, GPU, or FPGA) executing computer programs stored in memory. This module integrates a geometric feature calculation unit, a physical feature evaluation unit, and an optimization solution unit. The geometric feature calculation unit calculates features such as curvature and geodesic distance; the physical feature evaluation unit calls a simplified physical model to calculate preload stress; and the optimization solution unit executes batch joint optimization algorithms (such as maximizing the qNIPV acquisition function). The sampling module receives mesh data output from the data acquisition module, outputs a list of optimal sampling point coordinates, and sends it to the model building module. It should be understood that the sampling module can call multiple computing cores in parallel to accelerate the feature calculation and optimization search process on large-scale meshes.

[0089] The model building module is configured to build a proxy model based on the sampling point location. When the model parameter matrix becomes unstable, the kernel function is downgraded in a preset order until the value stabilizes.

[0090] The model building module is responsible for training and optimizing the parameters of the proxy model. This module receives the sampling point locations determined by the sampling module and calls a physical simulation interface (such as the FEniCSx solver or a simplified linear elastic solver) to obtain the corresponding stress tensor data as training labels. In terms of program implementation, this module includes a kernel function library, a matrix operation engine, and an anomaly handling unit. The kernel function library stores the definitions of various kernel functions, such as the Gibbs non-stationary kernel, the Matérn kernel, and the RBF kernel; the matrix operation engine is responsible for constructing and inverting the covariance matrix (e.g., Cholesky decomposition); and the anomaly handling unit monitors numerical stability in real time. Once ill-conditioned matrices are detected, the anomaly handling unit immediately triggers a degradation strategy, calling the next-level kernel function from the kernel function library to reconstruct the model. This modular design makes the expansion and replacement of kernel functions extremely convenient, enhancing the maintainability of the system.

[0091] The prediction module is configured to reconstruct the full-field stress field and quantify the prediction confidence level based on the surrogate model.

[0092] The prediction module is the output unit of the device. This module uses a surrogate model trained by the model building module to predict all unsampled points on the surface mesh of the target object. At the hardware level, this module can leverage the parallel computing power of the GPU to simultaneously predict the stress values ​​of tens of thousands of mesh points, achieving millisecond-level full-field reconstruction. The prediction module not only outputs the mean of the stress tensor but also calls the confidence quantization unit to calculate the upper bound of the generalization error based on PAC-Bayes theory, constructing a confidence interval. Finally, the prediction module packages and outputs the full-field stress data, uncertainty data, and confidence index to a display device for visualization (such as generating a stress contour map) or to a storage device for archiving.

[0093] Through the collaborative work of the aforementioned modules, the physics field reconstruction device in this embodiment achieves an automated process from data input to high-precision, robust prediction output. The modules communicate via standardized data interfaces, reducing coupling and facilitating system expansion. For example, when it is necessary to change the sampling strategy or kernel function type, only the internal logic of the corresponding module needs to be modified, without altering the overall architecture. This modular device design not only ensures the practical implementation of the method claims in engineering projects but also facilitates subsequent functional upgrades and maintenance.

[0094] Example 7: This example uses periodontal ligament stress field reconstruction as an example to describe in detail the specific application process and effects of the physical field reconstruction method described in the above examples. The periodontal ligament, as a typical thin soft tissue with a thickness of only 0.15 to 0.3 mm, is wrapped between the tooth root and alveolar bone and has a complex geometry, making it an ideal object for verifying the reconstruction capability of this invention under extremely sparse sampling conditions.

[0095] Obtain the surface mesh data of the target object.

[0096] In this embodiment, the input data is a three-dimensional surface mesh file of the periodontal ligament obtained through an oral scanning device, in STL format. In the example provided in this embodiment, the mesh model contains 12,158 mesh vertices, which discretely define the complex geometric manifold of the periodontal ligament. After loading the file, the system reads the three-dimensional coordinate information of all vertices and the connection relationships of triangular faces. To focus on mechanically sensitive areas, the system performs the region filtering operation described in Embodiment 5, setting a Z-axis threshold to eliminate extremely small regions at the root apex, ultimately selecting approximately 800 vertices as candidate sampling regions. This preprocessing step effectively reduces the search space of subsequent optimization algorithms and improves computational efficiency.

[0097] Based on the surface mesh data, a batch optimization sampling strategy is executed to determine the sampling point locations.

[0098] Specifically, this embodiment sets the sampling budget at 8 points, meaning only 8 miniature pressure sensors are allowed. From 800 candidate points, the system first calculates four-dimensional scores: curvature characteristics, geodesic distance characteristics, preload stress characteristics, and basic variance characteristics. Due to the extremely high curvature and stress concentration at the cervical margin of the periodontal ligament, candidate points in this region receive high geometric and physical pre-scores. Subsequently, the system performs batch joint optimization, using the posterior variance reduction as the acquisition function to jointly evaluate the combined effect of the 8 points. The optimization results show that the final 8 sampling points are not uniformly distributed, but adaptively clustered in key areas with large stress gradients, such as the high curvature area of ​​the cervical margin and the apical region, while being sparsely distributed in the gentler central area of ​​the periodontal ligament. This layout perfectly matches the mechanical properties of the periodontal ligament, avoiding waste of sensor resources.

[0099] Based on the location of the sampling points, a proxy model is constructed.

[0100] Specifically, the system invokes a three-layer adaptive physical simulation mechanism to obtain the true values ​​of the stress tensor at these eight sampling points. Since this embodiment runs on a high-performance computing workstation, the system successfully invokes the first-layer exact solver, calculating high-precision stress data based on the Ogden hyperelastic constitutive model. Subsequently, the system constructs a Gaussian process surrogate model based on a Gibbs non-stationary kernel function and integrates the results of the linear elastic finite element method as the physical prior mean. During model training, due to the clustering of sampling points at the cervical region, the local data density is extremely high, and the covariance matrix once exhibits an ill-conditioned warning. The system immediately triggers a kernel function degradation strategy, automatically downgrading the kernel function from a Gibbs non-stationary kernel to a Matérn 5 / 2 kernel, successfully ensuring the numerical stability of the model. This process verifies the engineering necessity of the degradation mechanism under extreme sampling conditions.

[0101] Based on the surrogate model, the full-field stress field is reconstructed and the prediction confidence is quantified.

[0102] This embodiment utilizes a trained surrogate model to predict stress at all vertices of the periodontal ligament surface mesh, outputting the three-dimensional stress tensor and principal stress amplitude for each vertex. Simultaneously, based on PAC-Bayes theory, the upper bound of the generalization error for each prediction point is calculated, constructing a confidence interval at a predetermined confidence level. To verify the reconstruction effect, this embodiment employs a comprehensive evaluation standard including the primary objective, constraint objectives, and confidence objectives. The primary objective is the ability to capture key regions, measured by Peak Stress Capture Rate (PSCR), Gradient Capture Rate (GCR), and sampling efficiency. PSCR is defined as the proportion of vertices in the top 10% of high-stress mesh vertices by principal stress amplitude that are no more than 2 mm away from any sampling point. GCR is defined as the proportion of vertices in the top 10% of high-gradient mesh vertices by stress gradient that are no more than 2 mm away from any sampling point. Sampling efficiency is defined as the ratio of the average stress at the sampling point to the global average stress. The constraint objective is that the reconstruction error in the global and high-stress regions is not inferior to the uniform sampling baseline. The confidence objective is to output confidence intervals and coverage statistics based on PAC-Bayes theory. For example, in comparison, under extremely sparse sampling conditions with 8 sampling points corresponding to 12158 grid vertices and a sampling ratio of approximately 0.066%, the global RMSE in this embodiment is 13.3693 Pa, compared to 13.3861 Pa for the uniform sampling baseline; the RMSE in the high-stress region is 18.6018 Pa, compared to 18.6168 Pa for the uniform sampling baseline, indicating that this method can maintain a reconstruction error level no worse than uniform sampling under extremely sparse sampling conditions. Simultaneously, the PSCR in this embodiment is improved from 3.7% to 51.4%, the GCR from 3.0% to 5.1%, and the sampling efficiency from 1.00x to 1.19x, indicating that the finite sampling budget is more optimally allocated to mechanically sensitive regions. The average coverage rate at a 95% confidence level, obtained based on PAC-Bayes theory, is 56.7%, providing traceable quantitative evidence of uncertainty in the prediction results.

[0103] This embodiment, using the periodontal ligament as a specific application scenario, fully demonstrates the entire process from STL file input to the output of the full-field stress cloud map and confidence index. Experimental data shows that the present invention can maintain a full-field reconstruction error that is basically equivalent to that of a uniform sampling baseline under extremely sparse conditions using only 8 sampling points and a sampling ratio of approximately 0.066%, and significantly improves the ability to capture high-stress and high-gradient regions, thereby enhancing the interpretability and application value of the physical field reconstruction results under sparse sampling conditions. It should be understood that although this embodiment uses the periodontal ligament as an example, this method is also applicable to the physical field reconstruction of other biological soft tissues (such as the cornea and blood vessel walls) or engineering components with thin-layer structures or complex geometries, such as aero-engine blades, thin-walled shells, composite laminates, flexible thin-film structures, pressure vessel walls, or other load-bearing components with complex curved surfaces. Only the corresponding physical simulation parameters need to be adjusted according to the geometric characteristics of the specific object.

[0104] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for reconstructing a physical field, characterized in that, include: Obtain the surface mesh data of the target object; Based on the surface mesh data, a batch optimization sampling strategy is executed to determine the sampling point locations; Based on the sampling point locations, a proxy model is constructed. When the model parameter matrix becomes numerically unstable, the kernel function is downgraded sequentially according to a preset order until the values ​​stabilize. Based on the surrogate model, the full-field stress field is reconstructed and the prediction confidence is quantified.

2. The method according to claim 1, characterized in that, The batch optimization sampling strategy includes: Based on the geometric and physical characteristics of the surface mesh data, a pre-score for the candidate sampling points is determined; Based on the pre-scoring, batch joint optimization is performed to determine the location of the sampling points.

3. The method according to claim 2, characterized in that, The geometric features include curvature features and geodesic distance features; the physical features include preload stress features and foundation variance features. The batch joint optimization uses the posterior variance reduction as the acquisition function.

4. The method according to claim 1, characterized in that, The surrogate model employs a non-stationary kernel function and incorporates physical prior means.

5. The method according to claim 4, characterized in that, The non-stationary kernel function is the Gibbs kernel function, and its length scale is adaptively determined by a neural network. The step of sequentially downgrading kernel functions according to a preset order includes: when the covariance matrix becomes ill-conditioned, downgrading from a non-stationary kernel function to a stationary kernel function, until the covariance matrix is ​​positive definite.

6. The method according to claim 1, characterized in that, The quantitative prediction confidence level includes: Based on the upper bound of the generalization error, the confidence interval of the prediction results is determined.

7. The method according to claim 6, characterized in that, The upper bound of the generalization error is determined in the following way: The upper bound of the generalization error is calculated based on the KL divergence between the posterior and prior distributions, the number of training samples, and the confidence parameter. The confidence interval is constructed based on the upper bound of the generalization error and the prediction mean.

8. The method according to claim 1, characterized in that, The acquisition of the surface mesh data of the target object includes: Load the 3D mesh file and filter the regions based on the coordinate range to obtain candidate sampling regions.

9. The method according to claim 1, characterized in that, The construction of the proxy model also includes: Obtain the stress tensor data at the sampling point location; The stress tensor data is obtained through a three-layer adaptive physical simulation, which includes an exact solution, an intermediate-level solution, and a fast fallback solution.

10. A physical field reconstruction device, characterized in that, include: The data acquisition module is configured to acquire the surface mesh data of the target object; The sampling module is configured to execute a batch optimization sampling strategy based on the surface mesh data to determine the sampling point locations; The model building module is configured to build a proxy model based on the sampling point location. When the model parameter matrix becomes unstable, the kernel function is downgraded in a preset order until the value is stable. The prediction module is configured to reconstruct the full-field stress field and quantify the prediction confidence based on the surrogate model.

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