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arxiv: 2604.27208 · v1 · submitted 2026-04-29 · 🧮 math.OC

Reliability-based Topology Optimization using Large Deviation Theory

Maryam Maghazeh , Ayyappan Unnikrishna Pillai , Mohammad Masiur Rahaman , Subhayan De This is my paper

Pith reviewed 2026-05-07 08:06 UTC · model grok-4.3

classification 🧮 math.OC
keywords reliability-based topology optimizationlarge deviation theorystochastic gradient descentfailure probability estimationstructural design under uncertaintyrobust optimizationcompliance and stress criteria
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The pith

Large deviation theory provides closed-form estimates of rare failure probabilities for efficient reliability-based topology optimization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to integrate large deviation theory with stochastic gradient descent to solve reliability-based topology optimization problems without the computational burden of repeated Monte Carlo simulations for small failure probabilities. It derives exponential rate functions that give both the probability estimates and their gradients directly from a few samples, allowing design updates in a mini-batch fashion. This matters because conventional approaches are too slow for realistic engineering structures, leaving designers to choose between slow reliable optimization or faster but less safe robust optimization. The method is demonstrated on beam benchmarks under compliance and stress constraints, where the optimized designs meet explicit reliability targets better than those from robust topology optimization alone.

Core claim

By applying large deviation theory, the framework obtains closed-form exponential rate estimates of rare event probabilities for structural failure criteria without parametric assumptions on the density. These estimates and their gradients are then used to drive mini-batch stochastic gradient descent updates of the topology design variables using only a few random samples per iteration, eliminating the need for a full nested reliability analysis loop at each optimization step.

What carries the argument

The large deviation rate function, which quantifies the exponential decay rate of the probability of rare failure events in the compliance or stress responses, serving as both the failure probability approximator and the source of gradients for the stochastic optimizer.

If this is right

  • RBTO designs achieve lower failure probabilities than robust topology optimization designs across the tested benchmarks.
  • Optimizing structural performance under uncertainty alone does not ensure satisfaction of explicit reliability constraints.
  • The approach scales to large problems because it requires only a few random samples per iteration instead of many nested simulations.
  • Direct applicability to safety-critical engineering tasks such as designing beams and cantilevers with compliance or stress limits.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar LDT-based rate estimates could be developed for other structural failure modes like buckling or vibration.
  • The framework might integrate with existing finite element software to enable practical adoption in industry design workflows.
  • Extensions could handle correlated uncertainties or non-Gaussian distributions by adjusting the rate function derivation.
  • Testing on more complex geometries would reveal if the computational savings hold beyond the simple beam cases.

Load-bearing premise

Large deviation theory supplies accurate closed-form exponential rate estimates and usable gradients for the compliance-based and stress-based failure criteria in the topology optimization setting without requiring parametric assumptions or full nested reliability loops at each iteration.

What would settle it

Running a high-sample Monte Carlo simulation on the final optimized designs from the benchmarks and comparing the observed failure probabilities to the LDT-predicted rates to check for close agreement.

Figures

Figures reproduced from arXiv: 2604.27208 by Ayyappan Unnikrishna Pillai, Maryam Maghazeh, Mohammad Masiur Rahaman, Subhayan De.

Figure 1
Figure 1. Figure 1: Most probable point ξ ∗ (ρ; z) from large deviation theory. decay. The importance of the rate function is that it simplifies the analysis of probabilities for rare events. Instead of calculating probabilities directly (which is often intractable for small probabilities), the rate function gives an asymptotic estimate. This property makes LDT a powerful tool for understanding systems with low-probability, h… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of the rectangular beam used in Example I. view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of optimized designs obtained from three approaches: (a) proposed LDT based RBTO, (b) robust view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of objective over the iterations in Example I. view at source ↗
Figure 5
Figure 5. Figure 5: Probability of failure estimated during the optimization along with the probability of failure of the final design view at source ↗
Figure 6
Figure 6. Figure 6: A two-dimensional L-shaped beam with uncertain material property and subjected to an uncertain load view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of optimized designs obtained from two approaches: (a) proposed LDT based RBTO and (b) view at source ↗
Figure 8
Figure 8. Figure 8: Evolution of objective over the iterations in Example II. view at source ↗
Figure 9
Figure 9. Figure 9: Probability of failure during optimization along with the probability of failure of the final design estimated view at source ↗
Figure 10
Figure 10. Figure 10: Typical stress distribution inside the L-shaped beam in Example II. view at source ↗
Figure 11
Figure 11. Figure 11: 3D cantilever beam used in Example III. 13 view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of optimized 3D cantilever beam using (a) the proposed RBTO approach and (b) robust design view at source ↗
Figure 13
Figure 13. Figure 13: Evolution of objective over the iterations in Example III. view at source ↗
Figure 14
Figure 14. Figure 14: Probability of failure during optimization along with the probability of failure of the final design estimated view at source ↗
read the original abstract

Reliability-based topology optimization (RBTO) requires repeated estimation of small failure probabilities and their gradients, making conventional nested Monte Carlo approaches computationally prohibitive for large scale structural problems. We propose an RBTO framework that integrates large deviation theory~(LDT) with stochastic gradient descent~(SGD) to address this challenge. LDT provides closed-form exponential rate estimates of rare event probabilities, enabling accurate gradient computation without parametric assumptions on the failure density and without evaluating a full nested reliability loop at every iteration. These LDT-based gradient estimates are used directly to drive a mini batch SGD update of the design variables using only a few random samples per iteration. The framework is validated on three benchmarks, namely, a two-dimentional (2D) simply supported rectangular beam, a 2D L-shaped beam, and a three-dimentional (3D) cantilever beam, under both compliance-based and stress-based failure criteria. Across these examples, the RBTO designs achieve failure probabilities lower than their robust topology optimization counterparts, demonstrating that optimizing for performance under uncertainty alone does not guarantee the satisfaction of explicit reliability constraints. Therefore, the proposed framework offers a computationally efficient route to reliable structural design under uncertainty, with direct relevance to safety-critical engineering applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a reliability-based topology optimization (RBTO) framework that integrates large deviation theory (LDT) to obtain closed-form exponential rate estimates for rare-event failure probabilities and their gradients, combined with mini-batch stochastic gradient descent (SGD) to update design variables using only a few samples per iteration. This avoids nested Monte Carlo loops. The approach is applied to compliance-based and stress-based failure criteria and validated on three benchmarks (2D simply supported beam, 2D L-shaped beam, 3D cantilever beam), where the resulting RBTO designs are reported to achieve lower failure probabilities than those obtained from robust topology optimization.

Significance. If the LDT rate functions and gradients remain accurate for the moderate failure probabilities and non-i.i.d. structural responses encountered in the benchmarks, the framework would provide a computationally scalable route to RBTO that is directly relevant to safety-critical applications. The demonstration that robust optimization alone does not guarantee explicit reliability targets is a useful clarification, and the avoidance of parametric assumptions on the failure density is a technical strength.

major comments (2)
  1. [Numerical results / benchmarks] In the numerical results section describing the three benchmarks, the central claim that RBTO designs achieve lower failure probabilities than robust TO counterparts is supported only by the LDT estimates themselves. No side-by-side comparison of these LDT estimates against direct Monte Carlo (or importance sampling) estimates on the final optimized topologies is reported. Because LDT is an asymptotic result, this check is required to confirm that the observed reliability improvement is not an artifact of the rate-function approximation or its gradient for the targeted probability range (10^{-2}–10^{-4}) and the compliance/stress response distributions.
  2. [Method / gradient computation] The gradient computation section (implicit differentiation of the rate-function optimization used inside the mini-batch SGD update) relies on the Gärtner–Ellis theorem supplying a differentiable rate function I(·) for the structural responses. The manuscript does not verify that the steepness and differentiability conditions hold for the finite-noise, non-i.i.d. compliance and stress random variables in the benchmarks; any violation would propagate bias into the design updates and undermine the reliability of the reported failure-probability reductions.
minor comments (2)
  1. [Abstract] Abstract contains spelling errors: 'two-dimentional' should read 'two-dimensional' and 'three-dimentional' should read 'three-dimensional'.
  2. [Method] The description of how mini-batch variance is controlled (or whether variance-reduction techniques are applied) in the SGD updates is brief; a short paragraph or pseudocode clarifying the sampling strategy would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback on our manuscript. We value the suggestions for strengthening the validation of our LDT-based approach. We address each major comment below and will incorporate the necessary revisions to improve the paper.

read point-by-point responses

  1. Referee: In the numerical results section describing the three benchmarks, the central claim that RBTO designs achieve lower failure probabilities than robust TO counterparts is supported only by the LDT estimates themselves. No side-by-side comparison of these LDT estimates against direct Monte Carlo (or importance sampling) estimates on the final optimized topologies is reported. Because LDT is an asymptotic result, this check is required to confirm that the observed reliability improvement is not an artifact of the rate-function approximation or its gradient for the targeted probability range (10^{-2}–10^{-4}) and the compliance/stress response distributions.

    Authors: We agree with the referee that an independent validation using direct Monte Carlo simulations on the final optimized designs is essential to substantiate the claims, given the asymptotic nature of LDT. In the revised manuscript, we will perform and report large-scale Monte Carlo simulations (using 10^5 or more samples) on the RBTO and robust TO topologies for all three benchmarks. This will allow a direct comparison of the failure probability estimates from LDT and MC, confirming the reliability improvements for the probability levels of interest. revision: yes

  2. Referee: The gradient computation section (implicit differentiation of the rate-function optimization used inside the mini-batch SGD update) relies on the Gärtner–Ellis theorem supplying a differentiable rate function I(·) for the structural responses. The manuscript does not verify that the steepness and differentiability conditions hold for the finite-noise, non-i.i.d. compliance and stress random variables in the benchmarks; any violation would propagate bias into the design updates and undermine the reliability of the reported failure-probability reductions.

    Authors: We acknowledge that the original manuscript lacks an explicit verification of the steepness and differentiability conditions required by the Gärtner–Ellis theorem for the non-i.i.d. structural responses in our benchmarks. To address this concern, we will add a new subsection in the methods or numerical results that numerically verifies these conditions. Specifically, we will compute the empirical rate functions from the sample data and demonstrate their convexity, steepness, and differentiability for the compliance and stress criteria used. This will provide evidence that the gradient computations are reliable for the finite-noise cases considered. revision: yes

Circularity Check

0 steps flagged

No circularity: LDT and SGD applied as external tools; results do not reduce to self-defined or fitted inputs

full rationale

The paper applies large deviation theory (an established result from probability theory) to obtain closed-form exponential rate estimates for failure probabilities and their gradients, then feeds those estimates into mini-batch SGD for design updates. This chain relies on standard external theorems (Gärtner–Ellis, etc.) and does not define the rate function I(·) in terms of the optimized topologies or vice versa. Validation consists of comparing RBTO designs against robust TO on three independent benchmark problems under compliance and stress criteria; the reported lower failure probabilities are empirical outcomes, not quantities that the paper’s own equations force by construction. No self-citations are invoked as load-bearing uniqueness theorems, no parameters are fitted to a subset and then relabeled as predictions, and no ansatz is smuggled via prior work. The derivation is therefore self-contained against external mathematical tools and benchmark data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of large deviation theory to provide closed-form estimates of failure probabilities and their gradients for structural failure criteria without parametric assumptions on the density.

axioms (1)
  • domain assumption Large deviation principle applies to the rare-event failure probabilities arising in the compliance and stress-based criteria for the topology optimization problems considered
    Framework depends on LDT supplying accurate exponential rate estimates usable for gradient computation in the structural setting.

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