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How AI Works | Gradient Accumulation — Microbatches, Effective Batch Size and Correct Optimizer Steps

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eduKate Asia

Did you know that a model can train with an effective batch of 128 examples even when its accelerator can hold only eight examples at a time? Gradient accumulation runs several forward-and-backward passes, adds their gradient contributions, and performs one optimiser update at the end of the group. The purpose is to make an update from more training examples without storing all their activations simultaneously. It is not a trick that creates free memory, and it does not make every training procedure mathematically identical to using a larger physical batch.

The practical recipe is to define the intended examples or tokens per update, weight each microbatch correctly, leave gradients in place between microbatches, and call the optimiser only at the accumulation boundary. Mixed-precision scaling, distributed synchronisation, clipping, scheduler steps, partial final groups and checkpoints must follow that same boundary. This guide works through the arithmetic, explains errors that silently change learning, and shows how to verify a training run instead of trusting a configuration label.

Contents

The three different batch sizes

A microbatch is the collection of examples processed in one forward-and-backward pass on one worker. A per-device physical batch is what the device processes at once; in a simple loop it equals the microbatch. An accumulation window comprises several microbatches whose gradients contribute to one update. In data-parallel training there may also be multiple workers, so the global effective batch is usually microbatch examples per worker times accumulation passes times number of participating workers. These quantities answer different questions: memory per pass, number of backward passes before updating, and number of examples represented by an update.

For example, eight examples fit on each of two workers, and each worker accumulates four passes before a synchronized update. The global effective batch is 8 × 2 × 4 = 64 examples. There are eight microbatch passes across the workers for each optimizer step, but the devices operate simultaneously, so four rounds of forward and backward work occur per worker. Calling that setup ‘batch size 8’ omits the optimization quantity; calling it ‘batch size 64’ can mislead somebody who needs to estimate GPU memory. Record all three numbers.

An ‘epoch’ is a pass over the training data under a declared sampler; it does not automatically mean the same number of optimizer steps in two experiments. A dataloader may count physical batches, a progress bar may count optimizer updates, and a scheduler may count either depending on how it is called. Before changing accumulation settings, write down which counter each component consumes. The wrong counter can change the learning-rate history even when the effective batch is correct.

Why summing gradients can reproduce a larger-batch update

Suppose the loss for example i is L_i and the desired batch objective is the arithmetic mean of N independent per-example losses. Differentiation is linear, so the gradient of their mean is the mean of their gradients: (g_1 + … + g_N) / N. If every microbatch contains exactly m examples and there are K microbatches, backpropagating each microbatch’s mean loss divided by K produces the same total gradient, apart from floating-point ordering. That is the mathematical reason accumulation works; it is not because the optimizer somehow remembers examples.

The equivalence assumes that the forward computation for one example does not depend on which other examples were in its batch, that the loss is reduced consistently, and that model parameters do not change between contributing microbatches. It also assumes a single optimizer step uses the combined gradient rather than a succession of small updates. Momentum, adaptive moments and weight decay behave differently if the optimizer is stepped K times because each step changes its state and often the parameters. Accumulation postpones these changes.

Floating-point summation order can create small numerical differences, so equality in real programs is usually checked with tolerances rather than bitwise identity. If a model uses operations that couple examples within a batch, such as training-mode batch normalization, microbatch processing can change activations before gradients are even aggregated. The honest claim is conditional gradient equivalence for appropriately separable losses, not universal full-training equivalence.

A fully solved scalar-gradient case

Consider a fictional linear model with one adjustable weight and eight invented training examples. Their individual gradients at the current weight are 2, 4, 6, 8, 1, 3, 5 and 7. A single true batch of all eight examples has gradient (2 + 4 + 6 + 8 + 1 + 3 + 5 + 7) / 8 = 36 / 8 = 4.5. With ordinary SGD, learning rate 0.1 and no momentum or weight decay, the weight change is negative 0.45. That provides an independent expected answer.

Now suppose the hardware fits just two examples. The four microbatch mean gradients are (2 + 4)/2 = 3, (6 + 8)/2 = 7, (1 + 3)/2 = 2 and (5 + 7)/2 = 6. Divide each mean by the accumulation count four before backward. The accumulated gradient is 3/4 + 7/4 + 2/4 + 6/4 = 18/4 = 4.5. One optimizer step again changes the weight by negative 0.45. The numerator can also be checked directly: microbatch sums 6 + 14 + 4 + 12 = 36.

If we forget the division, the accumulated gradient becomes 18, four times too large, and the same learning rate moves the weight by negative 1.8. If we call the optimizer after each microbatch, later gradients are evaluated at changed weights and there are four state updates rather than one. Neither mistake is ‘equivalent, just a little faster’. The first changes the gradient scale; the second changes the optimization algorithm. A unit test against this eight-example case detects both.

Mean versus sum loss reductions

Many loss functions offer a mean reduction, but ‘mean’ is not enough information. It might mean per example, per token, per valid pixel, per element or per sample-and-feature pair. With equal-sized microbatches and a genuinely per-example mean, dividing each microbatch loss by K is convenient. With unequal valid-unit counts, averaging microbatch means gives each microbatch equal influence rather than each valid unit equal influence. That difference is a statistical change, not merely a formatting issue.

Let microbatch A contain two valid units with summed losses 4 and 6, and microbatch B contain six valid units with losses summing to 12. The desired global mean is (4 + 6 + 12) / (2 + 6) = 22/8 = 2.75. The unweighted mean of the microbatch means is ((10/2) + (12/6))/2 = (5 + 2)/2 = 3.5. A model trained on the second objective gives unusually high weight to the smaller microbatch. Correct accumulation should use sums and divide by the total valid-unit denominator, or scale each microbatch mean by its count divided by the global count.

The key operational question is: what exactly is the denominator of the intended loss? If the application cares equally about each document, long documents should not automatically contribute more because they contain more tokens. If it cares equally about predicted tokens, the denominator is the number of non-masked predicted tokens. Neither policy is universally superior. Declare the policy, then implement and test it. Accumulation should not accidentally decide what the learning objective is.

The final window may be shorter

Assume there are ten physical batches in an epoch and the accumulation target is four microbatches per optimizer update. The natural groups have sizes four, four and two, so there are three completed optimizer steps if the final partial group is intentionally flushed. Dividing all ten microbatch losses by four makes the final gradient half the intended mean when each batch is equal sized, because only two contributions arrive. Dropping the last two batches instead changes training coverage. Neither policy should be an undocumented side effect.

A simple policy for equal-size batches is to scale losses in each final window by its actual number of contributing microbatches. For example, if the two remaining microbatch gradients are 6 and 10, their intended average is (6 + 10)/2 = 8. Dividing each by four would produce 6/4 + 10/4 = 4, which is wrong for an equal-weight two-batch update. More general implementations use the exact valid-unit counts, particularly when padding or variable lengths create unequal denominators.

An alternative is to carry accumulation state across nominal epoch boundaries, but then data shuffling, optimizer step timing, checkpoint semantics and reporting must explicitly support that choice. A framework may choose to flush on dataloader exhaustion; confirm its documented behaviour and test the final group with a tiny dataset. The last window is where many ‘almost correct’ training loops quietly diverge.

An update is not a backward pass

Training logs often say ‘step 100’ while hiding whether that means the hundredth microbatch or the hundredth optimizer update. If accumulation count is eight, 800 physical-batch forward-and-backward passes normally produce 100 optimizer updates, not 800. A scheduler configured for 800 steps but advanced only after updates experiences its milestones eight times more slowly than intended. The reverse error advances a schedule eight times in one effective batch, perhaps collapsing a warmup before useful weight updates occur.

Define explicit counters such as microbatch_index, completed_optimizer_updates, examples_seen and valid_tokens_seen. Increment the optimizer counter only when an optimizer step actually occurs. In mixed precision, a detected overflow can cause a step to be skipped; the policy for advancing the schedule then matters. Many pipelines advance only after a successful update to preserve a schedule tied to changed weights. Document the choice and verify the scheduler API rather than assuming its step name reveals the intended cadence.

Learning-rate and checkpoint articles on this site explore adjacent concerns: learning-rate schedules and resumption explains what a completed update means to a schedule. Gradient accumulation owns the different problem of forming that update from multiple passes. Linking the two is useful because a perfectly weighted gradient can still meet the wrong learning rate.

Zeroing gradients at the right boundary

In frameworks where backward adds to an existing gradient buffer, optimizer.zero_grad() removes the previous update’s gradient before a new accumulation window begins. Calling it after every microbatch destroys contributions from earlier microbatches. Calling it only at epoch boundaries carries a previous update’s residual gradients into the next group. Both mistakes can be silent when losses are decreasing for unrelated reasons.

A minimal conceptual loop is: clear gradients, for each microbatch compute its appropriately weighted loss and run backward, and at the end of the window perform validation of gradients, optionally clip, step the optimizer once, update the scaler if applicable, then clear gradients before the next window. The exact implementation depends on the framework, but the invariant is stable: only one optimizer update consumes the one combined intended gradient. A test that measures the gradient vector immediately before step is more informative than one that checks loss alone.

Some libraries set gradient buffers to None when zeroing, which can have performance and semantic differences for sparse or unused parameters. That detail should be handled according to the library contract. It does not alter the main point: gradient buffers are temporary accumulation state, and clearing them is part of the optimizer update protocol, not an arbitrary cleaning action.

Memory savings and their limits

Why does accumulation help a memory-constrained accelerator? A forward pass normally retains activations required for backward. Reducing microbatch size reduces many activation allocations, while parameter storage and optimizer state remain largely unchanged. Instead of holding activations for, say, 64 examples simultaneously, a device can repeatedly process eight examples, release the temporary graph after each backward, and keep only gradient buffers between passes. This often allows a larger effective update batch on available hardware.

It is not an eightfold reduction in total training memory when the physical batch falls from 64 to eight, because model parameters, momentum and adaptive optimizer state may dominate for large models. Nor does accumulation imply eightfold faster training; more separate forward/backward launches can increase overhead and lose throughput. Peak memory, total allocated memory, examples per second and optimizer updates per second are separate measurements. The right performance comparison reports all of them under identical sequence lengths and precision settings.

Activation checkpointing or recomputation saves memory by redoing some work rather than retaining every intermediate activation. Offloading moves state elsewhere. Distributed sharding partitions storage across devices. These techniques solve different constraints and can be combined with accumulation, but each changes performance tradeoffs. If the model itself cannot fit, shrinking the microbatch alone will not solve parameter-memory exhaustion.

Dropout and stochastic computation

Dropout introduces random masks during training. Under equivalent distributional assumptions it is possible for separately processed microbatches and a large physical batch to sample different random masks while still estimating the same expectation. They need not produce exactly the same gradient in one run. If strict reproducibility matters, record random number generator states and the order in which examples and operations are processed. Do not claim exact equality of trajectories merely because the expected batch size matches.

Stochastic augmentations create the same issue. An image rotation sampled once per epoch versus once per access can change gradients more than accumulation arithmetic. If different microbatch partitions produce different dropout masks or augmentation choices, a tolerant gradient comparison may need to disable stochastic transformations for the reference test. The purpose of such a test is to isolate scaling correctness, not to hide ordinary stochastic training variability.

One common debugging mistake is to compare the accumulated run and baseline after thousands of updates, notice diverged weights, and conclude that accumulation is broken. Chaotic optimization trajectories and different random streams make that evidence weak. Begin with one deterministic update on fixed inputs and fixed weights; then progressively restore dropout, mixed precision, multiple workers and the scheduler.

Batch normalization is not automatically equivalent

Training-mode batch normalization computes statistics from the examples processed together. A physical batch of 64 may use one mean and variance for 64 examples, while eight microbatches of eight use eight different means and variances. Even if their gradients are later averaged perfectly, the activations being differentiated are different. Running statistics may also receive several updates per effective batch rather than one, depending on the layer implementation.

This makes batch normalization a notable counterexample to the claim that microbatch accumulation reproduces large-batch training exactly. Normalization schemes operating independently for each example, such as many uses of layer normalization, avoid this particular cross-example coupling, though other sources of non-equivalence remain. It may be appropriate to freeze batch-normalization statistics or reconsider the architecture for small physical batches, but those choices themselves change the training procedure.

For a related deep dive, batch normalisation and train-eval statistics examines the statistics problem directly. This article concentrates on the accounting and update boundaries. Read both before interpreting a large-batch comparison as a clean optimizer experiment.

Automatic mixed precision and GradScaler boundaries

With float16 automatic mixed precision, some training loops scale the loss before backward to prevent tiny gradients from underflowing. The gradient scale must remain consistent throughout an accumulation window because gradients from earlier microbatches are already stored in scaled form. Changing the scale halfway through would combine numbers expressed in different units. A typical GradScaler checks for non-finite gradients and decides whether to skip an optimizer update at the full effective-batch boundary.

PyTorch’s official mixed-precision examples explicitly describe accumulation at effective-batch granularity, including scale updates and inf/NaN handling. Its guidance is specific to its APIs and should not be pasted blindly into other frameworks. The conceptual rule travels: if you are going to inspect, transform or clip a scaled gradient, first unscale once after all contributing backward passes are complete, and only then consume the gradient.

An optimizer update that is skipped due to non-finite gradients should be treated as a skipped update, not silently counted as a successful change in model weights. Log the scaler decision, gradient validity and completed optimizer updates separately from the count of attempted windows. This distinction also matters when resuming a run or judging progress from a learning-rate schedule.

Gradient clipping belongs after accumulation

Suppose two scalar microbatch gradients are +12 and -8 and the desired mean is +2. If a clipping threshold of 3 is applied separately to each raw microbatch before combining, the clipped contributions become +3 and -3, whose mean is zero. If the gradients are averaged first, the intended combined gradient is +2 and no clipping is needed. These are different algorithms. Clipping each microbatch individually can destroy cancellations that are legitimate properties of the full batch.

Combine appropriately weighted gradients first; unscale if using a loss scaler; then, if justified, apply a global-norm or other declared clipping method to the combined vector before stepping the optimizer. Some distributed implementations require a coordinated global norm rather than a per-device local norm. Review the framework’s order rather than inserting clipping wherever it is easiest to call a function.

The companion gradient clipping guide solves the norm calculation and explains why an overly small threshold can conceal an unstable underlying model. Accumulation and clipping are related but not interchangeable: one defines how contributions are combined, and the other may limit a completed contribution before an optimizer uses it.

Distributed training and synchronization

With synchronous data parallelism, each worker processes its own microbatches. At the update boundary their gradients must represent the intended global loss, often by averaging across workers. Synchronizing after every backward pass may be correct but wasteful; some frameworks support suppressing intermediate gradient all-reduces and performing the relevant reduction when the accumulated update is ready. That optimization saves communication, but it must not suppress the final synchronization.

The global effective example count in a simple equal-sized setup is per-worker microbatch size × accumulation count × worker count. If eight workers each process four examples over three passes, the update covers 8 × 4 × 3 = 96 examples. If one worker’s final microbatch is smaller, merely averaging worker means may no longer equal an example-weighted global mean. Track total valid denominators across workers and use them to weight contributions correctly.

Do not silently generalize this arithmetic to asynchronous parameter servers, pipeline-parallel schedules or unusual gradient reduction conventions. Those systems may accumulate, average or apply gradients differently. The engineering test is to compare a tiny multiworker update with a single-worker reference under controlled data and explicit reduction definitions. An impressive device count does not prove that the intended objective was optimized.

Variable-length text and masked targets

Sequence tasks make loss denominators especially consequential. Two microbatches with eight examples each can contain very different numbers of unmasked next-token predictions because their sequences have different lengths. If the objective weights each prediction token equally, a microbatch with 400 valid tokens should contribute four times the total influence of one with 100 valid tokens, all else equal. Dividing each microbatch’s mean loss by two incorrectly gives them equal aggregate weight.

For a fictional pair of windows, microbatch A has 100 valid targets and a sum of per-token losses of 300; microbatch B has 400 valid targets and a sum of 800. The desired token-weighted global loss is (300+800)/(100+400) = 1100/500 = 2.2. The simple average of microbatch means is (300/100 + 800/400)/2 = (3+2)/2 = 2.5. The difference, 0.3 loss units, is not a rounding error. The correct gradient weighting mirrors the declared global denominator.

When using a library that internally reduces token loss, inspect what happens to ignored padding targets and how loss scaling interacts with gradient accumulation. Never infer that all tokens count because tensor shapes match. Document valid token counts, reduction type and group boundaries in experiment metadata so a later reviewer can reconstruct the actual training objective.

Weight decay, clipping and optimizer state

Modern optimizers may maintain momentum, moving averages of squared gradients and parameter-wise state. With accumulation, all those states normally advance once per combined update. If you increase accumulation count without changing total examples per epoch, you reduce the number of optimizer steps per epoch. For a fixed number of epochs this also reduces the number of times step-coupled weight decay and learning-rate schedule operations occur. Comparing runs by epochs alone is therefore incomplete.

Consider a simple invented run with 1,024 examples, physical batch 8 and no dropped data. Without accumulation there are 128 optimizer updates per epoch. With accumulation 4 there are 32 updates per epoch; each update sees 32 examples. Over ten epochs, the counts are 1,280 versus 320. If the goal is to compare equivalent training exposure, both see the same number of examples; if the goal is to compare optimizer-update count, they are not equivalent. Neither comparison is automatically the ‘fair’ one for every research question.

Changing the global effective batch can justify retuning a learning rate, but no universal scaling rule guarantees improved accuracy. Begin with validated reference behaviour and use controlled evaluation on held-out data. Moving to a larger batch is a resource and optimization decision, not a guarantee of a better model.

Correct checkpoint and mid-window resumption

An optimizer checkpoint saved exactly after a completed update can include model parameters, optimizer state, scheduler state, mixed-precision scaler state and the data-ordering state needed to continue consistently. If a checkpoint is saved in the middle of an accumulation window, the current gradient buffers, number and weight of contributions, pending data position and related random states may also matter. Saving only the model weights at that moment loses the unfinished gradient sum.

A safe policy is often to checkpoint only at verified optimizer boundaries and mark the completed update number and examples or tokens seen. For expensive training that must resume within an accumulation window, document all in-flight state required to reproduce the pending update. Resuming a mid-window checkpoint and then zeroing gradients may silently discard work; retaining gradients without knowing the denominator may double-count an example.

An explicit replay decision also matters when data are streamed from an external loader. If a reader repeats the last microbatch after restart while previous gradients from that microbatch survive, the sample has twice the intended influence. Record stable sample or batch positions and test interrupted-and-resumed runs against uninterrupted baselines at least for a small deterministic model.

How to diagnose a silently wrong loop

If a model trains but loss is noisy, do not immediately blame the larger effective batch. Inspect one complete update boundary. Verify the same examples were processed as in the reference; the loss reduction was correct; gradients were not cleared between contributions; clipping occurred after combination; the optimizer stepped only once; and any scheduler or scaler advanced according to documented update rules. These are observable invariants rather than guesses from the final accuracy number.

Use a diagnostic with a tiny deterministic model and explicit per-example gradients. Save the expected combined gradient vector and compare it to the actual pre-step vector with a stated numerical tolerance. Then compare the weight change after one optimizer step. Repeat after enabling AMP and multiple workers separately. A failure at the gradient comparison points to weighting, missing contributions, clearing or synchronisation; a failure only after the step points more strongly to optimizer state or scheduling.

Useful logs are attempted_windows, successful_optimizer_updates, microbatches_contributed, valid_denominator, global_gradient_norm_before_clipping, scale_factor_if_used, skipped_updates and examples_seen. Avoid recording personal training samples or tokens merely for debugging. Aggregated counters and numerical summaries usually suffice. The aim is to make a wrong loop falsifiable without generating a privacy problem.

Fictional production case: fitting a student feedback classifier

Imagine a fictional educational research team training a text classifier on 12,800 consented, appropriately prepared examples. Its accelerator fits 16 examples at a time and the team wants effective batches of 128 examples. On one worker, accumulation count eight creates 128-example updates. The full dataset requires 12,800/128 = 100 completed updates per epoch if batching is exact and no examples are filtered. At physical batch 16 there are 800 backward passes per epoch. These two counters are both correct, but they describe different work.

Now suppose 64 examples are removed after an audit of duplicated records. The remaining 12,736 examples give 99 full 128-example updates and a final remainder of 64 examples: 99 × 128 = 12,672; 12,736 − 12,672 = 64. If the training policy flushes a half-size final update, it performs 100 updates. If it drops the final group, it trains on 12,672 examples. If it carries them into the next epoch, sample coverage and shuffling semantics must be explicit. No one should report 100 identical 128-example updates.

Success would mean that the run’s mathematics matches its policy, not that it attains a particular score. All figures here are invented to illustrate accounting. Real student work requires appropriate permissions, privacy controls and independent model evaluation. A small reproducible reference update should be proven before the pipeline is trusted to produce educational recommendations.

Changed-context transfer: uneven microbatches

Try a harder invented update window. A device is supposed to combine three microbatches whose valid example counts are 5, 5 and 2. Their per-example gradient sums for one parameter are 10, 20 and −6. What is the globally example-weighted gradient? Add the gradient sums to get 10 + 20 − 6 = 24, then divide by the actual total of 5 + 5 + 2 = 12 examples. The answer is 2. An ordinary SGD rate of 0.05 gives a weight change of negative 0.1.

Suppose instead we averaged microbatch means equally. Their means are 10/5 = 2, 20/5 = 4 and −6/2 = −3. The equal-mean average is (2 + 4 − 3)/3 = 1. That is half the intended gradient, leading to a negative 0.05 update rather than negative 0.1. The entire discrepancy came from a denominator that silently switched from examples to microbatches. The arithmetic is small enough to use as a reusable unit test.

The transfer rule is general: choose the fundamental unit of the objective before writing the accumulation loop. It may be examples, documents, pixels or valid tokens. Add contributions in compatible units and divide by the total intended denominator, including partial windows. When the training context changes, rerun the worked check rather than assuming the old ‘divide by K’ rule always applies.

Experimental comparison and meaningful limitations

A credible benchmark reports peak memory, throughput in examples or tokens per second, completed optimizer updates, evaluation results, failed or skipped updates and hardware configuration. Compare physical batch and accumulation policy separately. If you simultaneously change the learning rate, total training tokens, model architecture and optimizer, performance differences cannot be attributed cleanly to accumulation. A controlled experiment changes one relevant mechanism while holding others fixed or explicitly modelling their differences.

The greatest benefit of accumulation is fitting a desired global update batch on memory-limited devices. The main costs are additional forward/backward overhead, increased latency between updates, potentially different batch-dependent layer behaviour and greater bookkeeping burden. It cannot repair corrupt labels, compensate for unsuitable data splits or guarantee gradient stability. Reproducibility depends on sampler order, random operations, precision, distributed reductions and checkpoint practice.

For readers evaluating claims about training efficiency, remember that a technique can reduce peak memory without reducing computational work. Ask for the denominator in every comparison: per optimizer update, per example, per token, per epoch or per second. Without that denominator, even impressive percentages can be technically true but operationally misleading.

Frequently asked questions

Does accumulation change model accuracy? It can, because it changes update frequency and often the global batch size, and because batch-dependent layers or stochastic operations can differ. It does not have a single predictable direction of effect. How many accumulation passes should I use? Start from the desired global examples or valid tokens per update, divide by the feasible per-worker microbatch size and worker count where appropriate, then examine final partial groups.

Should I divide the loss by accumulation count? Only when that creates the intended weighting. For equal-sized microbatches whose losses are per-example means, often yes; for unequal valid counts, use total units. Should the scheduler step after every backward? Generally it should follow its explicitly defined unit, commonly completed optimizer updates. Does accumulation make training faster? It often saves peak activation memory relative to one large physical batch but can add overhead.

Can I combine accumulation with mixed precision and clipping? Yes, if scaler and clipping operations occur at the correct boundaries and the framework supports their order. Is accumulation equivalent to a big physical batch with batch normalization? Not automatically, because the forward statistics differ. What if the epoch ends mid-window? Flush, drop or carry according to an explicit policy; the last window’s denominator must match that policy.

Reader pathways and primary documentation

If you are a student, start with the eight-gradient arithmetic and the unequal-microbatch transfer test; these develop the algebra behind training. If you are an engineer implementing a loop, validate loss denominators, optimizer boundaries and one deterministic update before scaling out. If you review research, insist on effective-batch definitions, step counts and comparable total data exposure. These are complementary routes rather than an instruction to memorize every API.

Primary implementation guidance includes PyTorch’s automatic mixed precision examples, especially the gradient accumulation section, and Hugging Face Accelerate’s gradient accumulation guide for framework-managed boundaries. Confirm the versions and behaviour that match your actual environment. For nearby mechanisms, read batch normalisation and learning-rate schedules.

From the broader How AI Works series, learners can connect these calculations to model training and evaluation. The education and transfer guide explains why solving a familiar example is only the beginning: the harder test is recognizing which assumptions change when the setting changes. That habit is as valuable in a classroom as it is in model engineering.

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