Part III - Intelligent Simulation Selection and Multi-Fidelity Optimization

In the previous article, we developed the foundations of intelligent engineering search.

A Hybrid Neural Network-Gaussian Process surrogate was introduced to provide two critical pieces of information for every candidate design:

• an expected performance metric $\mu(\mathbf{x})$,
• an uncertainty estimate $\sigma(\mathbf{x})$.

This surrogate does not simply approximate the engineering response. It provides a probabilistic representation of the design space, allowing the optimisation framework to distinguish between regions of high predicted performance and regions of limited knowledge.

However, prediction alone is insufficient.

The surrogate must also determine where additional simulations provide the greatest engineering value.

This decision process is formalised through acquisition functions, which balance predicted performance, uncertainty, and information gain.

The next step is therefore to examine how these concepts are applied within an adaptive optimisation framework.

This transition—from simply asking "what is the best design?" to asking "what simulation provides the most value?"—is known as Active Learning.

Active Learning: Intelligent Simulation Selection

The hybrid surrogate model developed in the previous section fundamentally changes the information available to the optimisation algorithm. Rather than providing only an estimate of engineering performance, the surrogate now produces both a predicted response and an estimate of the confidence associated with that prediction.

Each candidate design therefore becomes

$$\mathbf{x} \rightarrow \left( \mu(\mathbf{x}), \sigma(\mathbf{x}) \right),$$

where:

• $\mu(\mathbf{x})$ is the predicted engineering performance,
• $\sigma(\mathbf{x})$ quantifies the uncertainty of that prediction.

This additional information transforms optimisation from a purely numerical search problem into a sequential decision-making process.

The objective is no longer to simply identify the design with the highest predicted performance.

Instead, the optimisation framework must determine which simulation will produce the greatest increase in engineering knowledge while respecting a finite computational budget.

---

Simulation as an Information Source

Traditional optimisation treats each simulation as an independent objective function evaluation.

Active learning adopts a different perspective.

A high-fidelity simulation is viewed as an expensive experiment whose primary purpose is to reduce uncertainty within the design space.

Consider the pulse jet example.

Suppose two candidate exhaust geometries are proposed.

The first lies within a region densely populated by previous simulations.

The surrogate predicts

$$F = 392 \pm 3 \text{ N}.$$

The second lies near the boundary of the explored design space.

The surrogate predicts

$$F = 430 \pm 95 \text{ N}.$$

A conventional optimisation algorithm immediately selects the second design because the predicted thrust is higher.

An active learning framework instead recognises that the second prediction is also substantially more uncertain.

The engineering value of simulating this design extends beyond determining whether the predicted thrust is correct.

Regardless of the outcome, the new simulation will significantly improve the surrogate model in a poorly understood region of the design space.

The simulation therefore generates both engineering performance data and information regarding the structure of the design space itself.

---

Exploration and Exploitation

This introduces one of the central trade-offs in sequential optimisation.

Every new simulation must balance two competing objectives.

Exploitation seeks designs that are expected to improve the current optimum.

These designs are selected because

$$\mu(\mathbf{x})$$

is large.

Conversely, exploration prioritises regions where the surrogate possesses limited knowledge.

These designs are selected because

$$\sigma(\mathbf{x})$$

is large.

An effective optimisation strategy must continuously balance these competing objectives.

Excessive exploitation causes the optimiser to repeatedly sample regions that are already well understood, producing only marginal improvements.

Excessive exploration wastes computational resources investigating regions that are unlikely to contain competitive designs.

The optimal strategy depends upon both the engineering objectives and the current state of knowledge represented by the surrogate.

---

Acquisition Functions

The acquisition function introduced in Part II provides the mathematical mechanism for this decision.

While the surrogate predicts the behaviour of the engineering system, the acquisition function evaluates the value of obtaining additional information. Different acquisition functions therefore represent different strategies for balancing exploration, exploitation and information gain.

Rather than asking

Which design appears best?

the acquisition function asks

Which simulation is expected to provide the greatest benefit?

Several acquisition functions have been developed for this purpose.

One of the simplest is the Upper Confidence Bound (UCB),

$$UCB(\mathbf{x}) = \mu(\mathbf{x}) + \beta \sigma(\mathbf{x}),$$

where

• $\mu(\mathbf{x})$ encourages exploitation,
• $\sigma(\mathbf{x})$ encourages exploration,
• $\beta$ determines the relative importance of uncertainty.

A larger value of $\beta$ causes the optimisation to investigate unexplored regions more aggressively.

As additional simulations reduce uncertainty throughout the design space, the optimisation naturally shifts towards exploitation.

Alternative acquisition strategies include Expected Improvement (EI), which estimates the expected improvement over the current best solution,

$$EI(\mathbf{x}) = \mathbb{E} \left[ \max \left( 0, f(\mathbf{x})-f^{*} \right) \right],$$

where

$$f^{*}$$

is the best design identified thus far.

Unlike UCB, Expected Improvement directly evaluates the potential benefit of a simulation relative to the current optimum, while still incorporating predictive uncertainty through the probabilistic surrogate.

The choice of acquisition function influences how aggressively the optimisation explores unknown regions, but the underlying principle remains the same.

The objective is not to optimise the surrogate directly.

It is to maximise the engineering value obtained from each expensive simulation.

---

Sequential Bayesian Optimisation Loop

Active learning operates as an iterative cycle.

At each step, the surrogate model represents the current knowledge of the design space.

The acquisition function then determines the next most valuable simulation:

$$\mathbf{x}_{next} = \arg\max_{\mathbf{x}} A(\mathbf{x})$$

The selected design is evaluated using the high-fidelity model:

$$\mathbf{y}_{next} = \mathbf{f}(\mathbf{x}_{next})$$

The new observation is added to the simulation database:

$$D_{k+1} = D_k \cup (\mathbf{x}_{next},\mathbf{y}_{next})$$

The surrogate is then retrained and the process repeats.

The optimisation therefore becomes a closed learning loop:

$$\text{Predict} \rightarrow \text{Select} \rightarrow \text{Simulate} \rightarrow \text{Update} \rightarrow \text{Repeat}$$

This sequential structure is the foundation of Bayesian optimisation.

---

Multi-Objective Active Learning

The pulse jet case study has deliberately focused on multiple competing objectives.

Increasing thrust generally increases thermal loading.

Reducing structural mass often decreases stiffness.

Improving combustion efficiency may alter acoustic behaviour.

Consequently, the optimisation problem cannot be expressed as a single objective.

Instead,

$$\mathbf{y} = \begin{bmatrix} F & \eta & T_{wall} & \sigma_{max} \end{bmatrix}^{T},$$

defines a vector of competing performance measures.

The optimisation therefore seeks a Pareto frontier rather than a single optimum.

Within this context, the acquisition function must also become multi-objective.

Rather than estimating the improvement of a single performance metric, it estimates the expected expansion of the Pareto frontier.

One commonly adopted approach is Expected Hypervolume Improvement (EHVI).

Instead of asking,

How much larger will thrust become?

EHVI asks,

How much will this simulation expand the achievable trade-space between all engineering objectives?

This distinction is particularly important in engineering design, where decision-making rarely depends upon a single performance metric.

The objective is to improve the quality of the entire design space rather than maximise one isolated quantity.

---

Learning Where Physics Matters

An important observation emerges from this formulation.

The optimisation algorithm is not directly searching for the best pulse jet.

Instead, it is identifying where additional physical information should be obtained.

Every high-fidelity simulation becomes a carefully selected numerical experiment.

Each experiment is chosen because it is expected to either:

• improve the current best design,
• reduce uncertainty within the surrogate,
• expand the Pareto frontier,
• reveal previously unknown engineering behaviour.

The optimiser therefore becomes an information allocation system rather than a design generator.

This distinction is subtle but significant.

The engineering knowledge remains embedded within the governing equations, the numerical solvers and the experimental validation discussed in Part 1.

Active learning simply determines where those expensive physical models should next be evaluated.

However, the framework developed so far still contains an important assumption.

Every selected experiment is implicitly assumed to have the same computational cost.

As established in Part 1, this assumption is rarely valid.

A zero-dimensional thermodynamic model may execute in milliseconds.

A quasi-one-dimensional model may require seconds.

A transient reacting CFD simulation may require many hours.

Selecting the next design point is therefore only part of the optimisation problem.

The framework must also determine which level of model fidelity is appropriate for evaluating that design.

Addressing this problem requires extending active learning beyond experiment selection towards intelligent allocation of computational resources across an entire hierarchy of engineering models.

Multi-Fidelity Optimisation: Allocating Computational Resource

The active learning framework developed thus far assumes that every experiment carries the same computational cost. Under this assumption, the optimisation problem is limited to determining which design should be simulated next.

Real engineering workflows rarely satisfy this assumption.

As established in Part 1, engineering analysis is typically performed using a hierarchy of mathematical models spanning several orders of magnitude in both computational cost and physical fidelity.

For the pulse jet, this hierarchy may be represented as:

Experimental Validation
          ▲
          │
High-Fidelity Coupled Multiphysics
          ▲
          │
Reduced-Order Physics Models
          ▲
          │
Analytical Models

Each level contributes different engineering information.

• Analytical models rapidly capture first-order behaviour and provide useful scaling relationships.
• Reduced-order models introduce additional physical domains while remaining computationally efficient.
• High-fidelity multiphysics simulations resolve the coupled interactions between fluid mechanics, combustion chemistry, structural mechanics, heat transfer and acoustics.
• Experimental validation ultimately determines whether the computational models accurately represent physical reality.

Consequently, every candidate design presents two independent decisions:

1. Which design should be investigated?
2. Which model should be used to investigate it?

Traditional active learning only addresses the first.

Engineering optimisation requires both.

---

Simulation Fidelity as a Decision Variable

Rather than representing a simulation simply as

$$\mathbf{x},$$

the optimisation problem becomes

$$(\mathbf{x},\mathcal{M}),$$

where

• $\mathbf{x}$ is the design vector.
• $\mathcal{M}$ is the selected modelling fidelity.

For the pulse jet,

$$\mathcal{M}\in \{ \text{Analytical}, \text{Reduced Order}, \text{High Fidelity}, \text{Experiment} \}.$$

The objective is no longer to minimise the number of simulations.

Instead, it becomes:

Maximise the engineering information obtained for a given computational budget.

This represents a fundamentally different optimisation problem.

Suppose two candidate designs exhibit similarly high uncertainty.

The first lies within a region where a reduced-order acoustic model is sufficient to resolve the uncertainty.

The second involves strong combustion instabilities requiring a fully coupled reacting CFD simulation.

Although both designs may provide similar information, their computational costs differ by several orders of magnitude.

An intelligent optimisation framework should recognise this distinction.

---

Computational Resource Allocation

The optimisation process may therefore be viewed as a sequential allocation of computational resources.

Each simulation consumes a finite computational cost,

$$C_i,$$

while producing an increase in engineering knowledge,

$$I_i.$$

Rather than maximising performance directly, the optimisation seeks to maximise the information gained per unit computational cost:

$$\max \sum_i \frac{I_i}{C_i}.$$

Although simplified, this expression captures the central objective of multi-fidelity optimisation.

High-fidelity simulations are not avoided because they are expensive.

They are reserved for situations where the expected increase in engineering knowledge justifies the computational investment.

Conversely, inexpensive reduced-order models can be employed wherever they provide sufficient information to guide the optimisation process.

This perspective treats computational budget as a limited engineering resource that must be allocated intelligently.

---

Reinforcement Learning

Determining the optimal allocation strategy is itself a sequential decision-making problem.

Every simulation changes the surrogate model, modifies the uncertainty distribution and alters which future simulations become valuable.

The quality of each decision therefore depends upon the entire optimisation history.

This can naturally be formulated as a Reinforcement Learning (RL) problem.

Unlike supervised learning, where the objective is to predict an output from historical data, reinforcement learning seeks a policy,

$$\pi(s),$$

that determines the best action for a given system state.

Within the optimisation framework, the RL agent does not control the underlying physics.

It controls the engineering workflow.

Its role is not to optimise the pulse jet directly.

Its role is to determine how computational effort should be distributed across the design space.

---

State Representation

The state of the optimisation process contains significantly more information than the current design alone.

A representative state vector may be written as

$$s_t= \{ \mathcal{D}, \hat{f}, U, B, P \},$$

where

• $\mathcal{D}$ is the accumulated simulation database.
• $\hat{f}$ is the current surrogate model.
• $U$ is the uncertainty distribution across the design space.
• $B$ is the remaining computational budget.
• $P$ represents the current Pareto frontier.

Rather than observing only the engineering system, the RL agent observes the state of the optimisation process itself.

Its decisions are therefore based upon both engineering performance and current knowledge.

---

Action Space

The action performed by the RL agent is considerably richer than selecting the next design.

Each action consists of

$$a_t=(\mathbf{x},\mathcal{M}),$$

where

• $\mathbf{x}$ specifies the next candidate design.
• $\mathcal{M}$ specifies the fidelity level used to evaluate that design.

For example, the policy may determine that:

• one region of the design space requires only a reduced-order thermodynamic simulation,
• another requires structural finite element analysis,
• a third justifies a fully coupled reacting CFD simulation,
• a final region has sufficiently high uncertainty to warrant experimental investigation.

The optimisation therefore becomes adaptive not only with respect to geometry, but also with respect to the physical models employed.

---

Reward Formulation

The design of the reward function ultimately determines the behaviour of the optimisation system.

A suitable reward should encourage:

• improvement of the Pareto frontier,
• reduction of predictive uncertainty,
• efficient use of computational resources,
• avoidance of redundant simulations.

Conceptually,

$$R = \alpha\Delta P + \beta\Delta U_red - \gamma C,$$

where

• $\Delta P$ measures improvement of the Pareto frontier,
• $\Delta U_red$ represents reduction in surrogate uncertainty,
• $C$ is the computational cost of the selected simulation,
• $\alpha$, $\beta$ and $\gamma$ determine the relative importance of each objective.

Unlike conventional optimisation, where reward is based solely upon engineering performance, this formulation recognises that information acquisition possesses intrinsic value.

A simulation that substantially improves the surrogate may be more valuable than another that produces only a marginal increase in thrust.

---

Beyond Optimisation

At this stage, the engineering workflow has evolved considerably from the conventional simulation process introduced in Part 1.

Rather than repeatedly evaluating candidate designs using a single numerical model, the framework now operates as an adaptive computational system.

The surrogate predicts engineering performance.

The Gaussian Process quantifies uncertainty.

The acquisition function identifies regions expected to provide valuable information.

The reinforcement learning agent allocates computational resources across multiple levels of model fidelity.

The high-fidelity simulations remain the authoritative source of engineering truth.

Taken together, these components form a closed optimisation loop that continuously improves both the surrogate model and the quality of the candidate designs.

---

Towards Intelligent Engineering Search

The engineering workflow has now evolved beyond conventional optimisation.

Simulation is no longer treated as a repeated evaluation step.

Instead:

• surrogate models provide rapid approximations,
• Gaussian Processes quantify predictive confidence,
• acquisition functions determine valuable simulations,
• active learning guides sequential exploration,
• multi-fidelity methods allocate computational effort,
• adaptive strategies determine where and how physics models should be evaluated.

The result is not an autonomous replacement for engineering analysis.

It is an intelligent computational framework that uses simulations more effectively.

However, this capability introduces a deeper question.

An optimisation algorithm can efficiently navigate a design space, but it cannot determine whether the assumptions defining that design space remain valid.

The final challenge is therefore not computational.

It is epistemic.

How can engineering systems determine when their own predictions should be trusted?