Mapping Uncertainty: A Direct Approach to Parameter Discovery | #Sciencefather #researchers #probability

🔬 Inverse Discovery of Hidden Probabilities

🎯 A Probability-Preservation-Based Subdomain Inverse Mapping Framework for Random Parameter Identification in Complex Systems

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In the realm of computational science and advanced engineering, accurately characterizing the stochastic nature of underlying system parameters is a formidable challenge — particularly when these parameters cannot be measured directly, but only inferred from observed system responses.

This pioneering framework introduces a probability-preserving, subdomain-based inverse mapping strategy, meticulously designed to reconstruct the input probabilistic structure from known output distributions.

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🧠 The Conceptual Breakthrough

At the heart of this methodology lies the principle of probability conservation, articulated through the foundational relation:

$$P_Y(y)\,dy = P_X(x)\,dx$$

This mathematical equivalence ensures that the density of the mapped input parameters aligns probabilistically with the observed output, enabling direct identification of random input distributions without resorting to iterative optimization or global inversion.

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🧩 Subdomain Inverse Mapping Strategy

Rather than performing a global inverse search across the entire high-dimensional space — which is computationally prohibitive — the input domain is decomposed into local subdomains. Within each subdomain:

• 🎯 A local inverse surrogate model is constructed

• 🔄 Probability preservation is enforced locally

• 🔗 The local mappings are integrated to construct the global inverse map

This divide-and-conquer paradigm enhances computational tractability, accuracy, and scalability, especially in nonlinear, multivariate systems.

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🚀 Distinctive Advantages

✅ Direct probabilistic reconstruction of random inputs
✅ Seamlessly handles non-Gaussian, nonlinear, and non-stationary systems
✅ Eliminates the need for global optimization loops
✅ Retains physical interpretability and mathematical consistency
✅ Adaptable to a wide array of domains: computational mechanics, structural dynamics, geophysics, biomedical modeling, and beyond

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⚙️ Workflow Overview

1. Generate response data via simulation or experimentation

2. Estimate output PDF using nonparametric methods (e.g., kernel density estimation)

3. Partition input domain into manageable subregions

4. Construct inverse mappings per subdomain

5. Apply the probability-preservation condition

6. Reconstruct full input distribution

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🌐 Applications & Impact

• 🛠️ Structural Health Monitoring: Identify degradation parameters from vibration responses

• 🌍 Geotechnical Inference: Estimate soil or rock properties from field measurements

• 🧬 Biomechanics: Characterize tissue variability using non-invasive data

• 🚀 Aerospace Engineering: Assess reliability margins under stochastic loading

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🧾 In Conclusion

This framework represents a paradigm shift in inverse uncertainty quantification — merging mathematical rigor with computational elegance. By preserving probability and localizing the inverse problem, it offers a scalable, interpretable, and accurate pathway to uncover the hidden distributions driving complex system behavior.

A new lens for inverse discovery.
A new era for uncertainty modeling. 🔍

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