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Meta-learning PINN loss functions

Apostolos F Psaros, Kenji Kawaguchi, George Em Karniadakis

arXiv:2107.05544v1cs.LG

TL;DR

PINN-based PDE solving requires choices across architectures, optimizers, learning-rate schedules, and other hyperparameters. This paper uses gradient-based meta-learning to discover shared offline PINN loss functions from parametrized PDE task distributions, achieving improved performance even on out-of-distribution tasks and compared with online loss adaptation.

  • Problem

    PINN-based PDE solving requires selecting the architecture, optimizer, learning-rate schedule, and other hyperparameters.

    Source: §1.1 · p. 1

  • Method

    The method uses gradient-based meta-learning to discover PINN loss functions from task distributions defined by parametrized PDEs.

    Source: §1.2 · p. 2

  • Results

    Shared offline-learned losses improved PINN performance even for out-of-distribution meta-testing and compared with online loss adaptation.

    Source: §1.2 · p. 3, §5 · p. 41

  • Takeaways & Limitations

    A shared offline-learned loss function can improve PINN performance across task distributions, including test tasks outside the meta-training distribution.

    Source: §5 · p. 41

    Results observed for examples in Sections 4.3–4.4 are not included in the paper.

    Source: §4.2 · p. 25

Abstract

from arXiv · show

We propose a meta-learning technique for offline discovery of physics-informed neural network (PINN) loss functions. We extend earlier works on meta-learning, and develop a gradient-based meta-learning algorithm for addressing diverse task distributions based on parametrized partial differential equations (PDEs) that are solved with PINNs. Furthermore, based on new theory we identify two desirable properties of meta-learned losses in PINN problems, which we enforce by proposing a new regularization method or using a specific parametrization of the loss function. In the computational examples, the meta-learned losses are employed at test time for addressing regression and PDE task distributions. Our results indicate that significant performance improvement can be achieved by using a shared-among-tasks offline-learned loss function even for out-of-distribution meta-testing. In this case, we solve for test tasks that do not belong to the task distribution used in meta-training, and we also employ PINN architectures that are different from the PINN architecture used in meta-training. To better understand the capabilities and limitations of the proposed method, we consider various parametrizations of the loss function and describe different algorithm design options and how they may affect meta-learning performance.

1. Introduction

The introduction frames PINN training as a challenging hyperparameter-selection problem and motivates learning loss functions that encode prior knowledge about parametrized physics.

1.1. Related work and motivation

  • PINNs solve PDE problems by embedding neural-network approximations into differential operators and minimizing composite residual, boundary, and initial-condition losses.Source: §1.1 · p. 1
  • PINN performance depends on architecture, optimization settings, and physics-specific choices such as residual-point numbers and locations, making training theoretically and practically challenging.Source: §1.1 · pp. 1–2
  • Because loss functions affect optimization convergence, minima, and neural-network loss landscapes, selecting MSE, MAE, or another loss is an additional PINN hyperparameter, with MSE commonly used by trial and error.Source: §1.1 · p. 2
  • An adaptive parametrized loss proposed in can include standard losses and improve performance online, but it raises training cost and does not exploit prior problem knowledge.Source: §1.1 · p. 2
  • The introduction therefore asks whether PINN loss functions can encode the underlying physics of a parametrized problem.Source: §1.1 · p. 2

PDE in a loss function optimized offline?

The paper introduces gradient-based meta-learning for offline discovery of PINN loss functions over parametrized PDE task distributions, with theoretical properties, regularization, and alternative parametrizations guiding the method. The learned losses are evaluated on unseen tasks and can significantly improve PINN performance, including against an online adaptive loss.

1.2. Overview of the proposed method

  • The proposed meta-learned loss significantly improves PINN performance, even compared with the online adaptive loss in, while shifting loss-training costs offline.At meta-test time, the learned loss is used to solve unseen tasks until convergence.Source: §1.2 · p. 3
  • The method discovers PINN loss functions offline with gradient-based meta-learning over diverse task distributions defined by parametrized PDEs.Meta-training samples PDE tasks, tracks loss-parameter dependence through a few-step PINN inner optimization, and updates those parameters using the MSE of the optimized PINN parameters.Source: §1.2 · p. 2, §1.3 · p. 3

1.3. Summary of innovative claims •

  • The paper proves two desirable properties of learned losses, proposes regularization to enforce them, and notes that the parametrization in guarantees both.It also extends prior loss-function meta-learning [27] through alternative parametrizations and algorithm-design options.Source: §1.3 · p. 3
  • The paper evaluates representative benchmarks to assess algorithm-design options, applicability, and limitations across diverse PDE task distributions.The experiments are presented in Section 4, while theoretical results and additional designs and results appear in later sections and appendices.Source: §1.3 · p. 3, §1.4 · p. 4

2. Preliminaries

The preliminaries describe PINNs as minimizing composite residual and data-misfit objectives for forward and inverse PDE problems, then explain how loss functions shape optimization gradients. They also motivate studying each loss term separately within standard neural-network training.

2.1. PINN solution technique overview

  • PINNs represent PDE solutions with neural networks trained by minimizing composite PDE, boundary, and initial residuals, with inverse problems adding solution-data misfit while learning λ.The forward formulation trains the solution approximation; the inverse formulation jointly trains it with the PDE parameters.Source: §2.1 · p. 4
  • The PINN objective is a weighted sum of f, boundary, initial-condition, and solution-data losses, but their optimal weights are unknown and often chosen equally or by trial and error.Each term averages a pointwise loss over its corresponding dataset; squared ℓ2 discrepancy yields MSE when averaged.Source: §2.1 · p. 5

2.2. Loss functions in neural network training

  • The section reduces the composite PINN objective to separate loss terms for analysis and reviews how first- and second-order optimizers use gradients, histories, or Hessians.SGD uses first-order information, adaptive methods incorporate update history, and Newton or BFGS methods additionally use exact or approximate Hessians.Source: §2.2 · p. 6
  • In neural-network training, the loss function controls how discrepancies affect the objective gradient, including the relative influence of dimensions in multi-dimensional predictions.Squared ℓ2 loss treats dimensions uniformly, whereas other losses can alter gradient contributions through their prediction derivatives.Source: §2.2 · p. 7

3. Meta-learning loss functions for PINNs

This section formulates PINN problems as related PDE task distributions and frames meta-learning as gradient-based bi-level optimization of shared hyperparameters. The approach uses task-average final performance to optimize parameters through differentiable inner PINN training.

3.1. Defining PDE task distributions

  • Related PDE tasks are parameterized by λ drawn from p(λ), enabling information and efficient hyperparameter choices learned from representative problems to transfer across the family.PINN performance depends on choices including architecture, activation, optimizer, and collocation-point count, which otherwise require repeated validation or tuning for novel PDEs.Source: §3.1 · pp. 7–8

3.2. Meta-learning as a bi-level minimization problem

  • Meta-learning optimizes hyperparameters η for average final performance across tasks by casting task-specific PINN training as the inner problem and validation performance as the outer objective.The resulting formulation alternates outer and inner optimization, with inner training solving the PINN objective until convergence and the outer objective measuring final validation error across tasks.Source: §3.2 · pp. 8–9
  • Gradient-based meta-learning computes total derivatives through the optimization path, requiring differentiable outer objectives and inner updates while allowing task resampling during meta-training.The paper limits discussion to gradient-based approaches and identifies exploding gradients from path differentiation as a relevant pathology.Source: §3.2 · pp. 9–10

13 end

This section presents a gradient-based method for meta-learning differentiable PINN losses, examining loss parametrizations and design choices including initialization, positivity constraints, and inner-path stability. It also develops remedies for exploding gradients and establishes desirable learned-loss conditions with theoretical guarantees and enforcement strategies.

3.3. An algorithm for meta-learning PINN loss functions

  • The parametrization framework supports FFN and adaptive losses, with η representing either network parameters or adaptive robustness and scale parameters.These options extend Algorithm 1 to differentiable loss families for PINN meta-learning.Source: §3.3 · p. 11
  • The algorithm trains PINNs with a parametrized loss ℓη, then updates η by differentiating validation MSE through the inner optimization path and averaging performance across tasks.Additional solution data may be used in the outer objective, even when inner training uses only PDE residual and boundary/initial-condition data.Source: §3.3 · p. 11

3.4.1.1 Adaptive loss function.

  • The adaptive loss parametrization uses shape α for outlier robustness and scale c for the near-zero quadratic bowl, with constrained reparameterizations preventing degenerate values.The direct loss can cause trivial online optimization of α, whereas the meta-learning setup’s distinct outer objective avoids those trivial solutions; Eq. (19) also satisfies the required training conditions without regularization.Source: §3.4.1.1 · p. 12

3.4.1.2 NN-parametrized loss function.

  • The NN-parametrized loss uses a two-hidden-layer ReLU FFN with softplus output, and its parameters can be initialized to approximate a known loss such as the squared ℓ2-norm.The softplus output does not affect optimization by enforcing positivity; it contributes through its nonlinearity.Source: §3.4.1.2 · p. 13

3.4.1.3 Meta-learning the composite objective function weights.

  • The composite objective weights for PDE residual, boundary-condition, and initial-condition terms can also be meta-learned as η, with softplus enforcing positive or minimum values.Different learned weight sets are equivalent up to an overall scaling factor.Source: §3.4.1.3 · p. 13

3.4.2. Inner optimization steps

  • Increasing the number J of inner steps exposes η optimization to more of the parameter-training history but increases computational cost and can cause exploding gradients.Approximations to the path derivative can reduce cost, while aligned η-gradients across steps can produce large Jacobians and unstable η updates.Source: §3.4.2 · p. 13
  • Experiments corroborate gradient clipping as the preferred stabilization method because dividing by inner iterations is too strict and norm normalization removes update-magnitude information.The η-gradient norm does not explode at every outer iteration, whereas normalization preserves direction but not magnitude.Source: §3.4.2 · p. 14

3.4.3. Theoretical derivation of desirable loss function properties

  • The optimal stationarity and MSE relation conditions enable efficient training, with the former ensuring global optimality and the latter ensuring a desirable stationary point exists.Theorem 1 and Theorem 2 establish global-minimum guarantees under the optimal stationarity condition, while Corollary 1 connects both conditions to global minimization of MSE under a rank condition.Source: §3.4.3 · pp. 14–16
  • Meta-learned losses may be non-convex yet remain tailored to the task distribution while satisfying the theoretically desirable conditions.Unlike standard PINN losses, meta-learning permits non-convex learned losses; standard PINN losses typically satisfy optimal stationarity because they are differentiable and convex in the prediction.Source: §3.4.3 · p. 15
  • The proposed penalty promotes zero first-order derivative at zero discrepancy and maximizes the derivative away from zero, while LAL automatically satisfies optimal stationarity and MSE relation conditions without regularization.Source: §3.4.3 · p. 16

4. Computational examples

This section evaluates meta-learned PINN losses across function approximation and PDE task distributions, parametrizations, optimizers, architectures, and data regimes. It examines in-distribution and out-of-distribution performance, generalization, objective-term adaptation, and the role of theory-driven regularization.

4.1. Discontinuous function approximation with varying frequencies and heteroscedastic noise

  • FFN and LAL parametrizations learned losses substantially different from MSE, with FFN producing more complex losses, especially in first-order derivatives.These differences were observed for MSE initialization during function approximation and in the corresponding loss-snapshot visualizations.Source: §4.1 · p. 18, Figure 2 · p. 19
  • Across 10 OOD function-approximation tasks, LAL losses achieved an average minimum relative test ℓ2 error at least 15% lower than all other considered loss functions.For example, LAL 4 achieved an average minimum rl2 error of approximately 17% near iteration 20,000.Source: §4.1 · p. 19, Figure 3 · p. 20

4.2. Task distributions defined based on advection equation with varying initial conditions and discontinuous solutions

  • For advection tasks with varying discontinuous initial conditions, SGD-based meta-training produced losses distinct from MSE, with FFN yielding more complex learned losses.The comparison included FFN and LAL parametrizations, with and without meta-learning objective-function weights, and Adam as an alternative inner optimizer.Source: §4.2 · p. 21
  • For SGD, objective-function-weight meta-learning initially improves performance but final learned losses combined with final weights eventually deteriorate it; all weights increase, while FFN and LAL balance initial conditions differently.For Adam, final learned losses remain close to MSE.Source: §4.2 · pp. 21, 23, Figure 5 · p. 23
  • Learned FFN and LAL losses achieve significantly smaller average minimum rl2 error than all considered standard and online adaptive losses on 5 ID tasks, despite only 20 meta-training iterations.Meta-testing uses SGD for 10,000 iterations; the comparison includes online adaptive losses.Source: Figure 8 · p. 25, §4.2 · p. 25
  • With Adam, learned FFN and LAL losses do not improve performance compared with MSE because Adam depends on gradient history extending beyond the 20 inner iterations used in meta-training.Higher Adam decay parameters, indicating greater dependence on the far past, further deteriorate learned-loss performance.Source: §4.2 · p. 26

4.3. Task distributions defined based on reaction-diffusion equation with varying source term

  • The reaction-diffusion tasks use a two-dimensional nonlinear steady-state equation with known boundary values and varying source terms generated from parameterized analytical solutions.The parameter vector λ controls amplitudes and frequencies in the constructed solution, from which zλ is obtained through the governing equation.Source: §4.3 · p. 27
  • Theory-driven gradient regularization corrected the FFN loss’s nonzero derivative-at-zero discrepancy and prevented the optimization divergence observed during meta-testing.Without regularization, the learned loss was shifted to the right and violated the MSE relation condition; the penalty of Eq. (28) fixed this issue.Source: §4.3 · p. 27, Figure 11 · p. 29
  • FFN learned losses achieved significantly smaller average minimum rl2 error than most standard losses across five OOD tasks, despite different meta-training tasks, architectures, and only 20 meta-training iterations.LAL did not generalize well, apparently because this example favored task-specific losses and its parametrization could not provide a sufficiently flexible shared loss.Source: §4.3 · p. 30, Figure 14 · p. 31

4.4. Task distributions defined based on Burgers equation with varying viscocity

  • In Burgers experiments, single-data losses had steeper derivatives because objective-function weights were absent, while meta-learning compensated across regimes by balancing PDE, BC, and IC terms.The two regimes produced nearly identical losses when weights were meta-learned, despite differing term balances; without weight meta-learning, their single-data losses differed slightly.Source: §4.4 · p. 32
  • Using analytical solution data in the outer objective produced highly different Burgers loss functions across the two regimes compared with single-data training.The corresponding loss-function snapshots were examined for both regimes and for single- and double-data cases using the FFN parametrization.Source: §4.4 · p. 32
  • Without regularization, the FFN loss trained with double data failed the optimal stationarity condition and caused divergence during full 20,000-iteration meta-testing despite satisfactory training and validation performance.Theory-driven regularization fixed this issue.Source: §4.4 · pp. 34, 38
  • Objective-function weights generally increased during meta-training, implying higher effective learning rates, while FFN and LAL parametrizations differed in how they balanced PDE, boundary-condition, and initial-condition residuals.Single-data losses had steeper derivatives because they lacked objective-function weights, which were greater than 1.Source: §4.4 · pp. 34–35, Figure 16 · p. 35
  • In regime r1, double-data training significantly reduced the outer objective and relative ℓ2 error, while the learned loss did not generalize well during full meta-testing for 20,000 iterations.The outer objective corresponds to loss after 20 iterations.Source: §4.4 · pp. 37–38
  • Learned FFN and LAL losses achieved significantly smaller average minimum relative ℓ2 error than most standard and online adaptive losses across five OOD tasks and both test regimes.The losses were meta-trained for only 20 iterations using a different PINN architecture and task distribution than those used at test time.Source: Figure 22 · p. 40, §4.4 · p. 40

5. Summary

The paper introduces offline meta-learning for PINN loss discovery across parametrized PDE task distributions, supported by theory on desirable loss properties. Experiments show shared learned losses can improve out-of-distribution performance, while Adam exposes generalization limitations.

Method

  • The method alternates task sampling, inner PINN optimization with the current learned loss, and outer updates based on final-parameter MSE.Meta-training continues until a stopping criterion is met while tracking learned-loss gradients.Source: §5 · p. 40

Theory and regularization

  • Two theorems establish that optimal stationarity can ensure gradient descent reaches a global minimum, and that combined stationarity and MSE relations make stationary points globally MSE-optimal.These results motivate a regularization method imposing the desirable conditions.Source: §5 · p. 41

Loss parametrizations

  • The learned adaptive loss (LAL), proposed in, automatically satisfies both conditions without regularization.This is one of the two loss parametrizations considered in the paper.Source: §5 · p. 41

Computational results

  • Shared offline-learned losses significantly improve performance under out-of-distribution meta-testing, including unseen task distributions and PINN architectures, and can outperform online loss adaptation proposed in.The evaluation covers regression, discontinuous approximation, advection, reaction-diffusion, and Burgers task distributions.Source: §5 · p. 41
  • Results demonstrate that loss parametrizations and other meta-learning algorithm design options materially affect performance across the evaluated task distributions.The study includes varying frequencies, initial conditions, discontinuities, source terms, and viscosity regimes.Source: §5 · p. 41

Limitations and future work

  • Adam remains a limitation: although LAL outperforms MSE for function approximation, neither FFN nor LAL generalizes satisfactorily in PINN examples, and higher β worsens performance.Adam’s dependence on exponentially decaying gradient history may explain this behavior; improving performance with memory-based optimizers is future work.Source: §5 · p. 41

Appendices · A. Loss functions with multi-dimensional inputs · B. Proofs

Appendix A develops increasingly expressive parametrizations for PINN loss functions with multi-dimensional inputs, extending directional one-dimensional losses toward vector-valued representations. The supplied passages do not provide substantive content for Appendix B’s proofs.

A. Loss functions with multi-dimensional inputs

  • Multi-dimensional PINN losses take two D_u-dimensional vectors as inputs and output a scalar measuring discrepancies among predicted and target quantities.The loss framework applies to PDE residuals, boundary conditions, and initial conditions.Source: A. Loss functions with… · p. 45
  • The baseline squared ℓ2 loss decomposes across output dimensions, with each directional term depending only on its corresponding prediction-target discrepancy.This construction reduces the multi-dimensional loss to a sum of one-dimensional losses.Source: A. Loss functions with… · p. 45
  • A first generalization replaces squared error with a parametrized function ˆℓη and weights each directional loss by a coefficient a_j.The weighted formulation is intended to make the learned loss more flexible and expressive.Source: A. Loss functions with… · p. 45
  • When combined with meta-learning, the weighted directional formulation can learn to normalize each directional loss optimally.This identifies a functional benefit of learning the coefficients jointly with the loss parametrization.Source: A. Loss functions with… · p. 45
  • Expressiveness increases by assigning different parametrized losses to different dimensions and by supplying both predictions and targets, rather than only their discrepancy.These variants allow dimension-specific and context-dependent loss behavior.Source: A. Loss functions with… · p. 45
  • The most general option replaces dimension-wise summation with a function that directly consumes the predicted and target vectors, yielding the highest expressiveness among the illustrated choices.Fig. A.1 orders the indicative representations by increasing expressiveness.Source: A. Loss functions with… · p. 45, Figure A.1 · p. 46

B.1. Proof of Theorem 1

The proof establishes that every stationary point of L is a global minimum, then characterizes when stationary points attain pointwise minima and constructs stationary points that need not do so.

B.1. Proof of Theorem 1

  • Every stationary point θ of L is shown to be a global minimum.The argument derives the stationarity conditions and uses the decomposition of L into constituent terms.Source: §B.1 · p. 47
  • Any θ whose predictions satisfy ∇qℓ(q, ui)|q=ûθ(xi)=0 for every i is a stationary point of L.This follows directly from Eq. (B.3) as summarized in the proof.Source: §B.1 · p. 47
  • If a global minimum of L exists, it achieves the global minimum value of ℓ(·, ui) for every training instance i.The conclusion uses the assumption that the prediction map spans R^(N D_u).Source: §B.1 · p. 47
  • Under the same spanning assumption, a non-minimizing stationary configuration can be constructed whenever ℓ has a non-global stationary point q.Specifically, q satisfies ∇qℓ(q, u)=0 while ℓ(q, u)>ℓ(q′, u) for some q′, and the construction realizes this condition across all i.Source: §B.1 · p. 48

B.2. Proof of Theorem 2

The proof establishes that the optimization trajectory has gradients converging to zero under both learning-rate cases. Theorem 1 then implies that any limit point is a global minimizer under the stated rank condition.

Proof of Theorem 2

  • In case (i), the loss sequence decreases monotonically, remains nonnegative, and converges, yielding the desired gradient-limit result.If the gradient reaches zero, it remains zero thereafter; otherwise, the decreasing bounded loss implies successive decreases vanish.Source: §B.2 · p. 49
  • In case (ii), the vanishing error term makes the loss eventually decrease, while a contradiction argument upgrades lim inf gradient norm zero to lim sup gradient norm zero.Summability arguments first establish lim inf_{r→∞}∥∇L(θ(r))∥ = 0; Lipschitz-gradient bounds then rule out a positive lim sup.Source: §B.2 · pp. 49–50
  • Under both learning-rate cases, lim_{r→∞}∇L(θ(r)) = 0, and Theorem 1 makes every limit point θ a global minimum when rank(∂û_X(θ)) satisfies the theorem’s condition.The proof treats vanishing gradients directly in case (i) and via a contradiction argument in case (ii).Source: §B.2 · pp. 49–50

B.3. Proof of Corollary 1

The proof shows that every stationary point of L interpolates all training data, so LMSE attains its global minimum there.

B.3. Proof of Corollary 1

  • The argument follows the proof of the first statement of Theorem 1, where q = ˆuθ(x_i) = 0 for all training points.Source: §B.3 · p. 51
  • At any stationary point θ of L, the assumption ∇qℓ(q, u) = 0 if and only if q = u implies ˆuθ(x_i) = u_i for every i.Source: §B.3 · p. 51
  • Consequently, LMSE(θ) equals its global minimum value at every stationary point of L.Source: §B.3 · p. 51

B.4. Proof of Proposition 1

The proof establishes both conditions of Proposition 1: ∇qℓ(q, u) = 0 if and only if q = u, and q = u satisfies optimal stationarity.

B.4. Proof of Proposition 1

  • The proof shows that ∇qℓ(q, u) = 0 if and only if q = u.The implication q − u = 0 ⇒ ∂ℓ(q,u) is derived using Eq. (B.42), and the converse is established by combining Eqs. (B.42) and (B.43).Source: §B.4 · p. 51
  • The proof establishes optimal stationarity by showing that q = u implies ℓ(q, u) ≤ ℓ(q′, u) for every q′ ∈ R.This inequality is obtained from the preceding bounds and Eq. (B.44), completing the optimal stationarity condition.Source: §B.4 · p. 51

C. Other algorithm design options

This section examines additional algorithm-design choices for task and neural-network resampling and for stopping meta-training. It also addresses metric noise by using long training and learned-loss snapshots in the computational examples.

C. Other algorithm design options

  • Resampling tasks and randomly initialized neural networks exposes the learned loss to broader task and initialization distributions but can destabilize training, depending on inner optimization steps.Tasks and initializations may instead be resampled every I′ and I′′ outer iterations rather than every iteration.Source: C. Other algorithm design options · p. 52
  • Stopping can use the outer objective or meta-validation error, but induced randomness may make either metric noisy, especially with T = 1.A less noisy metric or moving average can be used when appropriate.Source: C. Other algorithm design options · p. 52
  • The computational examples use 10,000 outer iterations, which works reasonably well for both metrics, and retain all learned-loss snapshots for meta-testing analysis.This strategy supports deeper understanding of the algorithm’s applicability to solving PDEs with PINNs.Source: C. Other algorithm design options · p. 52

D. Additional computational results related to the function approximation example

Additional function-approximation experiments show that exploding loss gradients can destabilize meta-training, while gradient clipping addresses this issue. They also show that more inner iterations improve learned-loss performance and robustness across design choices, with consistent patterns across test horizons.

Effect of loss function initialization

  • Without gradient clipping, the gradient explodes at outer iteration 4, causing a large jump in the loss-parameter norm.This behavior occurs with randomly initialized loss parameters and is shown in Fig. D.1.Source: Figure D.1 · p. 53, D. Additional computational results… · p. 53
  • The loss-gradient norm does not necessarily increase with more inner optimization steps, so subsequent experiments use gradient clipping rather than uniformly dividing gradients by the number of inner steps.The gradient behavior is evaluated as a function of both inner and outer iterations in Fig. D.2.Source: Figure D.2 · p. 53, D. Additional computational results… · p. 53

Design options experiment

  • Resampling tasks and re-initializing the approximator neural network introduce noise into meta-training, producing noisier outer-objective trajectories than non-resampling settings.The comparison is shown for design options varying inner steps, task-resampling frequency, and approximator re-initialization.Source: D. Additional computational results… · p. 54, Figure D.3 · p. 54
  • Increasing inner iterations improves learned-loss test performance and robustness across outer iterations and design options, with nearly identical patterns at 100 and 500 test iterations.This implies that selecting design options using either test horizon yields the same optimal choices.Source: D. Additional computational results… · p. 54, Figure D.4 · p. 55

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