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Residual Minimisation for Transport-Based Optimal Control

Tristan Pryer, Nikolaos Rekatsinas

arXiv:2610.02499v2math.NAcs.NA

TL;DR

The paper addresses boundary control for stationary photon transport, including angular-flux and dose-surrogate targets. It formulates a residual for the coupled KKT system and proves continuum error control, with neural-network and quadrature methods as an implementation. Numerical examples report small errors against exact optima, while weighted control residuals alone do not certify the objective norm.

  • Problem

    Radiotherapy transport control must choose an inflow source to match a target field or dose surrogate, while the control penalty and transport traces use different boundary measures.

    Source: §1 · p. 1

  • Method

    A residual least-squares functional encodes the transport KKT system, and neural networks minimize its quadrature approximation under analyzed approximation, quadrature, and optimization assumptions.

    Source: §1 · p. 1, §5 · p. 10

  • Results

    Manufactured dose tracking gives relative dose errors of 2.042 × 10^-4 in the disc and 5.625 × 10^-4 in the ball against exact optima.

    Source: §8 · p. 31

  • Takeaways & Limitations

    The continuum residual controls state and adjoint errors and the flux-weighted control error without a positive lower bound on |ω · n|; an unweighted optimality residual certifies control in the objective norm.

    Source: §1 · p. 1, §8 · p. 29

    Convergence of the weighted residual to zero need not imply convergence of the control in the unweighted objective norm, so the optimality residual must be strengthened for that guarantee.

    Source: §3 · p. 7

Abstract

from arXiv · show

We consider an optimal control problem governed by a stationary linear transport equation in two or three spatial dimensions, motivated by photon transport in radiotherapy treatment planning. The control is an inflow boundary source, and the observation operator may track either the angular flux or its angular integral, the latter serving as a simplified dose surrogate. The associated optimality system consists of a transport--adjoint pair with angular dependence, coupled through a boundary optimality condition. We formulate a residual least-squares functional whose minimisers coincide with solutions of the Karush--Kuhn--Tucker system and show that the functional controls the $L^2$ errors in the state and adjoint and the flux-weighted $L^2$ error in the control. This yields a continuum-level a posteriori estimate without recourse to discretisation. An additional unweighted optimality residual certifies the control in the norm of the objective. The estimate holds for every admissible approximation and accounts for the different boundary measures in the control penalty and transport traces, without a positive lower bound on $|\omega\cdot n|$. We then parameterise the state, adjoint and control using neural networks and minimise over a neural network space using appropriate quadrature. The Monte Carlo formulation replaces each residual integral by a weighted sample mean, whose quadrature error enters the a posteriori estimate. We prove convergence under explicit assumptions on approximation, quadrature and optimisation and derive bounds on the total error in terms of approximation, quadrature and optimisation error. Numerical examples in two and three dimensions illustrate the residual--error relation and the flexibility of the residual framework for field- and dose-tracking control problems, including regular manufactured solutions.

1. Introduction

The paper studies boundary control for transport models motivated by radiotherapy, where the target may be angular flux or its angular integral as a dose surrogate. It develops a continuum residual framework for the coupled optimality system and analyzes its neural-network and quadrature realization.

1. Introduction

  • The formulation isolates linear transport with pure attenuation and inflow control, tracking either angular flux or its angular integral as a simplified dose surrogate.Scattering, energy dependence, nonnegative delivery constraints, and more complete dose models are outside its scope.Source: §1 · p. 1
  • The residual least-squares functional encodes the full KKT system and controls state and adjoint errors in L2 and control error in the flux-weighted boundary norm, without excluding grazing directions.An additional unweighted optimality residual controls the control in the objective norm.Source: §1 · pp. 1–2
  • Neural networks approximate the state, adjoint, and control by minimizing a quadrature approximation of the residual, with convergence analyzed under explicit approximation, quadrature, and optimization assumptions.The continuum estimate also applies to other approximations when the residuals are well-defined.Source: §1 · pp. 1–2
  • The analysis addresses a coupled boundary-control system in which the unweighted control penalty meets flux-weighted transport traces, extending residual-error reasoning to the unknown inflow data.Related transport and control analyses provide relevant residual and trace frameworks, while the paper focuses on their coupling in the boundary optimality equation.Source: §1 · p. 2

2. Problem Setup

The problem setup defines a linear-quadratic transport boundary-control problem with either angular-flux or dose-surrogate tracking. Its transport spaces and optimality conditions yield a unique state–adjoint–control solution.

2. Problem Setup

  • The inflow flux is the control, and the objective balances target fidelity against an unweighted control cost while allowing angular-flux or angular-integral tracking.The angular integral is a simplified dose surrogate; controls have no sign constraint.Source: §2 · p. 3

2.1. Transport graph space and boundary trace.

  • The transport graph space requires finite inflow and outflow traces, while the objective uses an unweighted control space distinct from the flux-weighted trace spaces.The control space embeds continuously into the inflow trace space, but the converse bound fails near grazing directions.Source: §2.1 · p. 3

2.4. Lagrangian formulation and optimality conditions.

  • For every target and α > 0, the problem has a unique minimizer whose state, adjoint, and control solve the coupled KKT system.The boundary optimality condition couples the unweighted control penalty to the flux-weighted transport boundary term.Source: §2.4 · p. 5

3. Residual Stability

The residual functional characterises the unique KKT solution and gives a continuum stability estimate for admissible approximations. Weighted residuals do not alone certify control error in the objective norm, but an unweighted optimality residual can provide that stronger guarantee.

3. Residual Stability

  • The KKT solution is the unique global minimiser of Jb, with minimum zero; equivalently, Jb vanishes only at that solution.The residual domain can be enlarged to W × W × G− without changing the zero-residual solution.Source: §3 · p. 5
  • Every triple in W × W × G− satisfies a global residual-stability estimate under the stated hypotheses.The theorem’s constant depends on α, L, and ∥O∥.Source: §3 · p. 6
  • Unweighted boundary residuals also control unweighted control error, and fixed positive residual weights preserve stability with a modified constant.The modified constant is multiplied by 1 / min_i λ_i.Source: §3 · p. 7
  • A small weighted residual need not imply control convergence in U or closeness of the reduced objective to its minimum.The grazing-direction construction gives Jb tending to zero while the reduced-objective gap remains at least α/2.Source: §3 · p. 7
  • Measuring the optimality residual in U yields control accuracy in U; the combined-error constant scales as α^-2, while the objective-gap coefficient grows at most as α^-1.The stronger control guarantee requires no additional unweighted traces of the state or adjoint.Source: §3 · pp. 7–8
  • For angular-flux tracking, the characteristic formula gives a separate raywise control minimisation; dose tracking couples angular directions and lacks that separate minimisation.For nonnegative target data, the angular-flux tracking control given by the formula is nonnegative.Source: §3 · p. 8

4. Neural Network Approximation and Residual Minimisation

The paper approximates the transport optimality system by jointly trained neural networks and minimises a quadrature approximation of its residual functional. Deterministic or Monte Carlo quadrature supplies the loss, while quadrature error must be controlled to infer an error bound from the discrete loss.

4. Neural Network Approximation and Residual Minimisation

  • Smooth neural networks approximate the state, adjoint, and inflow control jointly, with their outputs inserted into the residual functional for training.The networks represent spatial and angular dependence and satisfy the regularity needed for transport derivatives and boundary traces.Source: §4 · pp. 8–9, §4.1 · p. 9

4.2. Loss Functional and Quadrature Discretisation.

  • The discrete loss uses weighted quadrature sums, while Monte Carlo replaces residual integrals with weighted sample means that are unbiased for each fixed admissible approximation.The boundary weights reflect the measures used by the selected residual functional.Source: §4.2 · p. 9, Monte Carlo loss · p. 9

4.3. Training Algorithm.

  • Training minimises the chosen quadrature loss jointly over network parameters, using samples fixed before optimisation for Monte Carlo quadrature.The procedure uses Adam and, where specified, continues with L-BFGS.Source: §4.3 · p. 10
  • The continuum stability estimate applies to every returned network, but a discrete-loss error bound also requires control of quadrature error.For Monte Carlo quadrature, this contribution is sampling error.Source: §4.3 · p. 10

5. Quadrature Error and A Posteriori Control

The section carries quadrature discrepancy into residual-based error control, distinguishing fixed-field Monte Carlo estimates from uniform control needed for sample-selected networks. It also treats deterministic rules and the extra inner angular quadrature used for dose tracking.

5.1. A posteriori residual control.

  • For every admissible approximation, residual stability turns the continuum residual—and a discrete loss with bounded quadrature discrepancy—into a solution-error bound; training-node loss alone is not a continuum certificate.The reliability constants can be conservative.Source: §5.1 · p. 11

5.3. Monte Carlo quadrature.

  • For a fixed field independent of samples, Monte Carlo loss is unbiased and has root-mean-square quadrature error O(M^-1/2) when squared residual densities have finite second moments.Bounded residual densities also support a high-probability estimate.Source: §5.3 · p. 12
  • Uniform quadrature control over a compact bounded network class covers sample-selected approximations with high probability, but required bounds and covering numbers may grow with architecture.The result is conditional; the implemented optimisation trajectories are not projected onto the prescribed compact parameter set.Source: Uniform control on the training class · p. 12
  • Independent fresh-sample validation can certify a frozen network under conditional residual bounds, but this route was not used for the reported trained networks.Reevaluation on a finer deterministic rule alone does not provide this probabilistic certificate.Source: Independent validation · p. 13

5.7. Deterministic quadrature.

  • Positive deterministic rules exact on constants bound each residual-integral error by its best approximation from the rule's polynomial space.Gauss–Legendre and spherical-design exactness yield convergence through approximation of the complete residual densities; smooth activations alone do not ensure exponential convergence.Source: §5.7 · pp. 13–14

5.9. Angular quadrature for dose tracking.

  • For dose tracking, inner angular quadrature adds a consistency term to the outer quadrature discrepancy; an unbiased estimate of Du alone does not make the squared adjoint residual unbiased.A uniform training estimate also requires this consistency term to be bounded uniformly over the network class.Source: §5.9 · p. 14

6. Neural Network Approximation and Convergence

Section 6 combines neural-network approximation with quadrature estimates to characterize convergence through approximation, quadrature, and optimization errors. Under its stated conditions, when all three contributions vanish, the state, adjoint, and control converge to the KKT solution.

6.1. Approximation in the graph and trace norm.

  • Under the stated closure and network-class assumptions, neural classes approximate the solution triple in X and their infimum residual tends to zero.The classes must eventually contain every three-output tanh network with two hidden layers and finite width and parameters.Source: §6.1 · p. 14
  • A sufficient approximation condition requires additional spatial H1 regularity, not guaranteed by well-posedness for arbitrary L2 data; it is not established for indicator-target examples.The sufficient condition requires no angular derivatives.Source: §6.1 · p. 14, §6.3 · p. 16
  • The approximation result uses increasing network classes and does not establish consistency for a single fixed architecture.It implies convergence for continuum near-minimisers whose optimization gaps tend to zero.Source: §6.1 · p. 15

6.3. Convergence.

  • When approximation, quadrature, and optimization contributions all vanish, the state, adjoint, and control converge to the KKT solution in the error norm.The convergence estimate separates these three contributions.Source: §6.3 · p. 16
  • The total-error estimate separates approximation, quadrature, and optimization, but the optimization gap remains an assumption rather than a guarantee from field-level convexity.The empirical nonnegative loss is an upper bound for the optimization gap; quadrature is used at both the computed network and the continuum comparison function.Source: §6.3 · p. 16
  • With δN→0, convergence is in probability; if ∑N δN<∞, convergence is almost sure.These conclusions hold under the stated random-quadrature hypotheses.Source: §6.3 · p. 17
  • On the quadrature-good event, E≤Cstab(aN+2qN+oN), with the factor two arising from quadrature at both comparison functions.Here aN, qN, and oN bound approximation, quadrature, and optimization contributions, respectively.Source: §6.3 · pp. 16–17
  • An M^-1/2 quadrature contribution to squared error yields an M^-1/4 solution-norm contribution; accuracy-cost bounds also require approximation rates and optimization-cost bounds.Source: §6.3 · p. 17

7. Numerical Results

The numerical results test neural-network approximations of field- and dose-tracking optimality systems against exact or manufactured solutions, examining residual–error behavior in two and three dimensions. They also explore joint dose-target approximation with one network, including ellipsoidal targets and connected bilobed targets with varying lobe separation.

Computational setting description.

  • Computations use fully connected networks for state, adjoint, and control, with residuals minimized on training quadrature and errors and final residuals evaluated on finer rules.Standard training uses 500 Adam updates followed by 750 L-BFGS updates; parametric ellipsoidal and bilobed cases use 2,000 outer L-BFGS updates.Source: Computational setting description · p. 17

7.1. Example: Field and dose tracking of a circular target in the square.

7.3. Example: Manufactured-solution verification of three-dimensional dose control.

  • In three-dimensional manufactured dose control, the final error-to-residual ratios are 0.5126 and 0.7281, and computed dose profiles closely follow the exact optimum despite its nonzero target mismatch.The final residuals are Jb,h = 5.2662 × 10−4 on the training rule and 5.2701 × 10−4 on the finer rule; the finer-rule unweighted residual is JU,h = 5.8803 × 10−4.Source: Table 8 · p. 25, §7.3 · pp. 25–26

7.4. Example: Parametric ellipsoidal dose targets in the unit cube.

  • One network jointly approximates five fixed-volume ellipsoidal targets; at the final update, the joint target mismatch divided by the known target norm is 0.3638.Training and independent-rule residuals are 5.4667 × 10−4 and 1.0393 × 10−2, respectively; these are target mismatches, not errors against an exact optimal dose.Source: §7.4 · p. 27

7.5. Example: Parametric bilobed dose targets.

  • One network is trained on five connected bilobed targets with fixed-size lobes and varying separation, by minimizing the arithmetic mean of their discrete residual functionals.The targets use µ ∈ {−0.5, 0, 0.5, 0.75, 1}; the lobe sizes remain fixed while target volume varies with separation, and all targets lie strictly inside the cube.Source: §7.5 · pp. 27–28
  • The joint relative target mismatch is 0.3373, while the recorded training residual decreases by approximately 64% between updates 1,400 and 2,500.The approximation is reported at update 2,500; the residual at that update is 7.6780 × 10^-4.Source: §7.5 · p. 29
  • The dose profiles follow the changing lobe separation with smooth boundary transitions and an interior overshoot; the most separated target's off-centre profile resolves the central dip, but positive dose remains in the gap.The off-centre profile is at t = 0.12.Source: §7.5 · p. 29, Figure 10 · p. 29

8. Conclusions

The paper establishes continuum residual stability for transport boundary-control optimality systems and extends the certification to the unweighted control norm. Numerical examples illustrate the residual analysis, while the conclusion identifies limits in regularisation and model scope.

8. Conclusions

  • The continuum residual bounds state and adjoint errors in the volume norm and control error in the flux-weighted boundary norm, without requiring a positive lower bound on |ω · n|.The result holds for fixed α > 0 and does not assert uniform stability as α → 0.Source: §8 · p. 29
  • An unweighted optimality residual certifies control error in the objective norm, but weighted residual convergence alone does not provide that stronger guarantee.The disc benchmark reports an unweighted relative control error of 1.043 × 10−3 on separate finer quadrature.Source: §8 · p. 30
  • Monte Carlo quadrature yields an M^−1/2 fixed-field loss-error rate under stated moment assumptions, while training-selected fields require uniform estimates over the neural-network class.The sampling rate concerns the loss and does not by itself establish the same rate for the solution norm.Source: §8 · p. 30
  • The manufactured dose examples report relative dose errors of 2.042 × 10−4 in the disc and 5.625 × 10−4 in the ball against their exact optima.The examples also report manufactured state-error results in two and three dimensions and assess residual–error correspondence at saved training stages.Source: §8 · pp. 30–31
  • Reliability constants may be large at small regularisation parameters, and the computations do not constitute a controlled study of sensitivity to α.For angular-flux tracking, the squared-error constant of the unscaled residual has an unavoidable α^−2 dependence.Source: §8 · p. 31
  • Extending dose tracking toward treatment planning requires nonnegative inflow constraints, while scattering and energy dependence are further model extensions.Nonnegative inflow constraints lead to a variational inequality requiring a corresponding change to the residual analysis.Source: §8 · p. 31

A.1. Explicit stability constants.

The appendix gives explicit residual reliability bounds, examines their dependence on regularisation and residual scaling, and describes sampling conditions for an unweighted optimality residual. It also identifies the scope of the supplementary experiments.

A.1. Explicit stability constants.

  • The matrix-residual bound provides an admissible reliability constant and can serve as an estimator using individual continuum residual norms.The factor two in the stated constant accounts for the factor 1/2 in Jb.Source: §A.1 · p. 31

A.2. Dependence on the regularisation parameter.

  • For angular-flux tracking, the sharpened constant uses κ = α^−1, but the unscaled residual's squared-error reliability constant has unavoidable α^−2 growth.The tabulated quantities are continuum bounds for flux-weighted control error, not fitted constants or observed errors.Source: §A.2 · p. 32, Table 11 · p. 32

A.3. Residual weights and reliability.

  • Rescaling the optimality residual preserves the zero-residual solution but changes the loss and reliability constant; a smaller constant alone does not establish a better bound or improved optimisation conditioning.The general reliability bound may be conservative for small α, and the residual-to-error ratio observed at one approximation does not determine a global constant.Source: §A.3 · p. 32

A.4. Sampling the unweighted optimality residual.

  • Sampling the unweighted optimality residual requires checking finite-variance and bounded-density conditions, since a density proportional to |r3|^2/b can have divergent second moment near grazing directions.Under the corresponding moment and uniform-class hypotheses, the quadrature proofs apply to JU.Source: §A.4 · p. 33

Appendix B. Supplementary numerical experiments

  • The supplementary experiments add a manufactured dose benchmark, target-control comparisons, and a fixed-field Monte Carlo quadrature test.Their computational settings are reported in Table 1.Source: Appendix B · p. 33

B.1. Triangular field target.

The triangular-target example compares a learned field with its characteristic reference and examines how target geometry appears in directional profiles and angular integrals. Evaluation reports separate state, control, and angular-integral errors.

B.1. Triangular field target.

  • The example uses a triangular target with σ = 0.1 and α = 10^−4, and evaluates the angular integrals using the same rule for approximate and reference maps.The state and control reference follow from analytic intersections of each ray with the target's three edges.Source: §B.1 · p. 33, Table 12 · p. 33
  • Directional profiles reflect the triangular target's direction-dependent projections and chord lengths, while the angular integral retains its geometry with transport smoothing.The training loss decreases to Jb,h = 1.499 × 10^−4.Source: §B.1 · p. 33
  • The relative state error is 3.281 × 10^−2, the relative control error is 3.601 × 10^−2, and the relative angular-integral error is 8.581 × 10^−3.The angular-integral maxima on evaluation nodes are 2.2323 for the approximation and 2.2429 for the reference.Source: §B.1 · p. 33

B.2. Example: Manufactured dose-tracking optimum.

This example constructs an exact dose-tracking optimum to separate numerical recovery error from the mismatch between the optimal dose and prescribed target. The learned solution recovers the optimum, with the reported checks made on a finer deterministic quadrature.

B.2. Example: Manufactured dose-tracking optimum.

  • The constructed fields satisfy the full dose-tracking KKT system without volume forcing, making the manufactured solution the unique optimum for its target.The construction uses α = σ = c = 1, and its target mismatch is c√π.Source: §B.2 · pp. 34–35
  • The finer-rule residual is 2.530 × 10^−5, and the unweighted squared optimality residual is 8.006 × 10^−6, giving JU,h = 2.825 × 10^−5.The training-rule residual is 2.502 × 10^−5.Source: Recovery of the optimal dose · p. 35, Table 13 · p. 35
  • The reported ratios are final-checkpoint observations, not estimates of uniform reliability constants, and saved loss history does not establish residual–error correspondence throughout optimisation.The dose error compares the learned angular integral with the exact angular integral and includes numerical angular integration of the learned field.Source: Recovery of the optimal dose · p. 35
  • The absolute dose error is 7.438 × 10^−4, while the target mismatch is 1.772440, verifying recovery of the dose-tracking optimum at α = 1.The target mismatch is close to √π ≃ 1.772454 and is intrinsic to the optimum, not a discretisation or optimisation error.Source: Recovery of the optimal dose · p. 35, §B.2 · p. 35

B.3. Example: Field and dose tracking of a circular target.

Field and dose tracking use the same transport system and prescribed indicator but observe different quantities, so their optimal states need not have the same angular integral. The dose-tracking solution has a smooth transition across the target boundary and a relative target mismatch of 0.27574.

B.3. Example: Field and dose tracking of a circular target.

  • The problems share the transport equation and boundary optimality condition, but field tracking uses source u − q while dose tracking uses D(u) − q.Both prescribe the same indicator, and the field target is unscaled.Source: §B.3 · p. 36
  • Dose tracking couples directions through D(u) − q, whereas the field-tracking optimum follows from (21); their directional fields are shown at 84° and 132°.The figure uses a common colour scale in each row, with the dashed circle marking ∂T.Source: §B.3 · p. 36, Figure 14 · p. 37
  • Dose tracking has a relative target mismatch of 0.27574, with a smooth transition across the target boundary; its evaluation-rule residual is mainly due to the adjoint component.The adjoint source contains the discontinuous target.Source: §B.3 · p. 36
  • Table 14 reports relative target mismatch, control norm, and residuals for the respective KKT systems, using a separate evaluation rule for the control norm and target mismatch.The mismatch is defined as the L2 norm of Dh(uθ) − q divided by the L2 norm of q.Source: Table 14 · p. 37

B.4. Residual approximation under Monte Carlo quadrature.

A fixed-field experiment isolates Monte Carlo quadrature error in the residual functional, while the analysis distinguishes this from sampling during network training. Across the tested sample sizes, the measured error follows the predicted M^-1/2 dependence.

B.4. Residual approximation under Monte Carlo quadrature.

  • The fixed fields perturb the Example 7.2 solution by ε = 0.05, scaling the residual functional by ε^2 while leaving the target unchanged.The interior and inflow residuals are specified explicitly, and the outflow residual is ε.Source: §B.4 · pp. 37–38
  • Each sample size uses independent uniform interior samples and independent inflow samples weighted by b, with the constant outflow contribution integrated exactly.The study repeats each size 256 times; smaller sample sets are prefixes of the same sample pools.Source: §B.4 · p. 38
  • The residual-estimation error fell from 6.89 × 10^-3 to 2.28 × 10^-4, with fitted exponent 0.492, agreeing with M^-1/2 for fixed fields.The experiment isolates quadrature error because the residual density and its variance remain unchanged as M increases.Source: §B.4 · p. 38, Figure 16 · p. 39
  • During network training, fields depend on the samples, so Proposition 5.6's uniform estimate is needed to control that dependence.The fixed-field experiment directly tests Lemma 5.4; repeated trials estimate mean-square sampling error, which need not be visible in one realization.Source: §B.4 · p. 38

Appendix C. Proofs of the Monte Carlo quadrature estimates

The Monte Carlo quadrature proofs establish error bounds using independent, centered samples for each residual term. They allow sample sharing across residuals and do not require independence between different residual terms.

Appendix C. Proofs of the Monte Carlo quadrature estimates

  • The proof frames each residual estimate through its integral and sample error, with the sampling measures and residual densities defined in Subsection 5.3.The stated Lipschitz bound applies for ∥θ − θℓ∥ ≤ η, almost surely.Source: Appendix C · pp. 39–40
  • Independence and centering for each fixed residual term, followed by Minkowski’s inequality, establish (26) without independence between residual terms.Samples may be shared across different residual terms.Source: Appendix C · p. 39
  • Hoeffding’s inequality bounds each sampling deviation when its threshold scale Bk is positive; terms with Bk = 0 vanish.The thresholds yield (29), and substitution into (24) gives (28).Source: Appendix C · pp. 39–40

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