---
license: cc-by-4.0
task_categories:
- time-series-forecasting
- other
tags:
- mechanics
- finite-element-analysis
- constitutive-modeling
- plasticity
- out-of-distribution
- physics-informed-machine-learning
pretty_name: AgentFEM Material Loading Memory
size_categories:
- 1K<n<10K
---

# AgentFEM Material Loading Memory

![Material loading-memory hysteresis](artifacts/t2_material_loading_memory_v1/hysteresis_preview.png)

An open, reproducible research dataset for path-dependent material modeling,
neural constitutive surrogates and finite-element deployment tests. The
repository contains six staged, explicitly separated releases.

## Start here

| Goal | Recommended entry |
|---|---|
| Understand the current dataset | This page and the [sealed-test data card](data/t2_denim_sealed_test_v1/README.md) |
| Train or benchmark a constitutive model | [DENIM start guide](DENIM_START_HERE.md) |
| Reproduce the multiaxial baseline study | [Multiaxial v2 guide](T2_V2_START_HERE.md) |
| Reproduce the original loading-memory release | [Original v1 guide](START_HERE.md) |
| Use the latest DENIM checkpoint | [AgentFEM-DENIM](https://huggingface.co/HaomingLuo/AgentFEM-DENIM) |

Current trajectory accounting:

| Role | Trajectories | Interpretation |
|---|---:|---|
| Training | 1,808 | Optimization data for the current v2.2 model |
| Validation | 503 | Checkpoint selection and anti-forgetting guardrails |
| Development test | 1,349 | Frozen historical benchmarks; later aggregate results informed subsequent model design |
| Prospective sealed test | 128 | Published after model freeze; never used for training or selection |
| **Total** | **3,788** | Complete histories, not exchangeable independent experiments |

## What is a loading history, and why does it matter?

A loading history is the ordered sequence of deformation applied to a material.
For an elastic spring, only the present deformation matters. For an
elastoplastic metal, the route matters as well: loading, unloading, reversing,
rotating the loading direction or adding a mean strain changes the material's
internal state. Two specimens can therefore arrive at the same current strain
with different stresses because they arrived there by different routes.

This dataset varies the history deliberately rather than merely sampling more
points on one curve:

| History variation | Question being tested |
|---|---|
| Proportional and reversed cycles | Can the model reproduce yielding, unloading and the Bauschinger effect? |
| Non-proportional and rotating multiaxial paths | Can it track changing stress directions and coupled material memory? |
| Nested minor loops and non-periodic sequences | Can it remember partial unload/reload events instead of resetting its state? |
| Amplitude and mean shifts | Can it extrapolate when the load becomes larger or biased to one side? |
| 100--200-cycle histories | Does a small state error accumulate into long-term drift? |
| Matched coarse/fine discretizations | Is the prediction tied to the physical path or to a particular step size? |
| Material-parameter variation | Can one conditioned model represent a family of materials rather than one fixed curve? |

The scientific object is therefore not an isolated stress value. It is the
mapping from **material parameters + ordered loading history + current internal
state** to the next stress and state. Randomly shuffling time steps or mixing
related resolution groups across data roles would destroy this meaning.

## Recommended evaluation metrics

Report absolute stress RMSE in MPa together with a scale-free metric. MPa is
needed for engineering interpretation; a relative metric makes results easier
to compare across cohorts and stress levels.

- **Relative L2 error (%)** = `||stress_pred - stress_ref||2 / ||stress_ref||2 × 100`.
  This is the preferred scale-free trajectory/state metric for signed cyclic
  stresses.
- **Yield-normalized RMSE (%)** = `RMSE / reference_yield_stress × 100`.
  This answers how large the average error is relative to the 280 MPa reference
  yield stress used by the sealed protocol.
- **R2** is useful as a supplementary global fit indicator, but should not
  replace an engineering error in MPa.
- **MAPE is not recommended** because cyclic stress repeatedly crosses zero;
  division by values near zero can make an accurate prediction look arbitrarily
  poor.

For the sealed reference data, the stress RMS scales are 359.540 MPa for global
path OOD, 91.411 MPa for 150/200-cycle histories, 371.687 MPa for amplitude and
mean shifts, and 206.814 MPa over all 128 trajectories. These denominators are
published so every model can be compared under the same convention. Model
scores remain in the model repository to keep this page focused on the data and
evaluation contract.

## Releases

- **T2 v1:** 1,008 proportional cyclic J2 trajectories for loading-memory
  baselines and forward/reversed ordering controls.
- **T2 multiaxial v2:** 1,024 independently generated J2 and Chaboche
  trajectories in the complete five-dimensional deviatoric strain space,
  with ID, path-OOD and parameter-OOD splits.
- **T2 DENIM closure v1:** 128 multiaxial trajectories from a deliberately
  richer three-memory, tabulated-hardening reference material. The published
  two-memory DENIM model must learn the missing internal-variable closure.
- **T2 DENIM boundary v1:** 500 additional trajectories for training-density,
  path, amplitude, long-history and integration-resolution boundaries.
- **T2 DENIM capability v4:** 1,000 trajectories for 100-cycle horizons,
  full six-component loading, AgentFEM-derived structural paths and paired
  discretization/noisy-observation tests.
- **T2 DENIM sealed test v1:** 128 prospectively generated trajectories for
  globally unseen paths, 150/200-cycle horizons and combined amplitude/mean
  shift. The DENIM v2.2 weights and evaluation protocol were frozen first.

The original 2,660-role protocol remains frozen; capability v4 adds its own
roles and the sealed release adds no optimization data. The six releases form
a staged evidence chain and must not be pooled or randomly reshuffled without
respecting material, path-family, resolution-group and release boundaries.

Historical `Path-OOD` labels are protocol-relative: they mean unseen for the
model and training split defined at that release. Some historical test-family
names entered later training releases, so those scores are not global OOD
claims for v2.2. `sealed_test_v1` is the current prospective global path-OOD
protocol.


## Historical material-conditioned DENIM v2 protocol

All 2,660 trajectories now have a frozen role in the conditional study: 1,484
train, 407 validation and 769 test. The release adds no duplicate data. Shared
known-material pretraining plus incomplete-material replay reduces long-history
RMSE from 10.906 to 4.263 MPa. Reproduction code, exact accounting and complete
metrics are in `conditional_v2/`.

## Historical fixed-material DENIM closure

![DENIM closure summary](artifacts/t2_graybox_closure_v1/closure_summary.png)

DENIM stands for **Discrete-Energy Neural Internal-variable Model**. The
reference material contains three kinematic memory channels and a non-
exponential tabulated isotropic-hardening curve. DENIM retains only two memory
channels and receives none of the reference hardening equations or parameters.

Held-out non-proportional path results:

| Model | Test RMSE | Test R2 |
|---|---:|---:|
| Incomplete J2, no learned hardening | 59.541 MPa | 0.730208 |
| GRU | 76.988 MPa | 0.548933 |
| **DENIM** | **1.136 MPa** | **0.999902** |

The 918-parameter DENIM also passed coarse/fine increment checks and three
notched-bar deployment gates. Cyclic and monotonic reaction relative-L2 errors
were 0.636% and 0.500%; the severe cyclic stress test required one global
trust-region fallback. This remains a fixed synthetic material study with
internal-state supervision, not an experimental calibration or a certified
production material.

The standalone weights and model card are published at
[HaomingLuo/AgentFEM-DENIM](https://huggingface.co/HaomingLuo/AgentFEM-DENIM).

## DENIM capability-boundary extension

![DENIM boundary comparison](artifacts/t2_denim_boundary_v1/boundary_model_comparison.png)

The 500-trajectory extension contains 250 training, 50 validation, 80 held-out
path, 60 amplitude-extrapolation, 40 long-history and 20 matched-resolution
histories.  No failed case was silently removed.

| Test | Incomplete J2 | GRU | Frozen DENIM | Expanded DENIM |
|---|---:|---:|---:|---:|
| Published held-out paths | 59.541 | 98.260 | 1.136 | **0.714** |
| New path OOD | 58.177 | 96.959 | 1.163 | **0.743** |
| Amplitude OOD | 69.315 | 95.896 | 3.091 | **2.294** |
| Long-history stress test | 54.394 | 90.757 | 11.261 | **10.906** |

Values are stress RMSE in MPa.  The long-history result is reported as a
current capability boundary, not hidden by the stronger ordinary path results.
Expanded-DENIM RMSE on matched 61/121/481/961-state histories was
0.673/0.722/0.761/0.767 MPa.

- Viewer-friendly index: `data/t2_denim_boundary_v1/index.csv`
- Lossless trajectories: `data/t2_denim_boundary_v1/cohort.h5`
- Data card: `data/t2_denim_boundary_v1/README.md`
- Quality report: `artifacts/t2_denim_boundary_v1/QUALITY_REPORT.md`
- Full metrics: `artifacts/t2_denim_boundary_v1/model_metrics.json`
- All-data usage: `docs/T2_ALL_DATA_STAGE_SUMMARY.md`


## DENIM capability extension v4

![Capability extension preview](artifacts/t2_denim_capability_v4/capability_extension_preview.png)

The v4 increment adds 1,000 complete trajectories: 300 long-cycle, 300 full
six-component non-proportional, 200 AgentFEM-derived local structural paths and
200 matched discretization/observation trajectories. The 100-cycle histories
contain 2,001 accepted material states. Fifty continuous paths are integrated
at 61, 121, 241 and 481 states and carry explicit group IDs; noisy and sparse
observations are fields paired with clean truth, not separately counted cases.

The structure-derived cohort uses local strain bases from twelve AgentFEM
linear-elastic unit-load solves on four perforated/notched plates, followed by
nonlinear reference-material replay. It does not claim coupled elastoplastic
redistribution. See `data/t2_denim_capability_v4/README.md` for the exact
protocol and `artifacts/t2_denim_capability_v4/capability_summary.json` for
machine-readable boundary statistics.

## Prospective sealed test v1

The 128-trajectory sealed test is a final, non-training extension: 64 globally
unseen path-shape histories, 32 histories with 150 or 200 cycles, and 32
combined amplitude/mean-shift histories. All eight path-family identifiers are
disjoint from training. The reference integrator enforces finite values,
monotone accumulated plastic strain, a relative yield-surface residual below
`1e-8`, total equivalent strain no greater than `0.0155`, and accumulated
equivalent plastic strain no greater than `0.10`.

The manifest binds the data to DENIM v2.2 weights SHA-256
`d7b848f41aa46616c76c5f0db536d8ecfbc508a5375faa3cbbcc555fb062633b`.
These samples are ineligible for training, validation, checkpoint selection or
model redesign. The repository-level protocol audit reports zero exact
cross-role collisions for full model inputs and for input/target pairs. This
dataset release intentionally contains no sealed-test model score; evaluation
is published separately in the model repository.

- Data card: `data/t2_denim_sealed_test_v1/README.md`
- Lossless trajectories: `data/t2_denim_sealed_test_v1/cohort.h5`
- Viewer indexes: `data/t2_denim_sealed_test_v1/index.csv` and `index.jsonl`
- Generation contract: `configs/t2_denim_sealed_test_v1.json`
- Leakage audit: `artifacts/t2_denim_protocol_audit_v1.json`

## AgentFEM runtime validation

The expanded DENIM checkpoint is also available as a checksum-authenticated
`safetensors` bundle in the
[AgentFEM-DENIM model repository](https://huggingface.co/HaomingLuo/AgentFEM-DENIM).
The safe bundle reproduces the legacy implementation on the fixed 121-step
path to a maximum stress difference of `2.68e-7 Pa` and identical final PEEQ.

Serial and two-rank AgentFEM implicit plastic-bar runs both completed 4/4
increments, with a reported maximum-stress difference of `5.96e-8 Pa` between
the two executions. The current plastic automatic-differentiation tangent has
a documented 0.05–0.80% discrepancy against fixed-old-state finite
differences over the audited plastic states. The demonstrated global cases
converge; an exact consistent plastic tangent remains outside the present
claim boundary.

- Machine evidence: `artifacts/t2_denim_agentfem_v1/runtime_validation.json`
- Validation note: `artifacts/t2_denim_agentfem_v1/RUNTIME_VALIDATION.md`
- Data/model handoff: `docs/T2_DATA_MODEL_HANDOFF.md`

## Multiaxial v2 at a glance

- eight path families, including non-proportional, rotating and random
  five-direction loading;
- 241 ordered states per trajectory;
- stress, strain, plastic strain, PEEQ, two-family total backstress, plastic
  increments and physical diagnostics;
- zero quality-gate failures across all 1,024 trajectories;
- six model families evaluated under three frozen protocols;
- element-level deployment in a controlled notched-bar finite-element gate.

![Multiaxial hysteresis examples](artifacts/t2_multiaxial_ood_v2/multiaxial_hysteresis_preview.png)

Stress RMSE on held-out trajectories:

| Model | ID | Path OOD | Parameter OOD |
|---|---:|---:|---:|
| Pointwise MLP | 83.51 MPa | 105.81 MPa | 82.34 MPa |
| GRU | 17.26 MPa | 76.03 MPa | 19.26 MPa |
| LSTM | 22.67 MPa | 102.54 MPa | 22.89 MPa |
| Causal TCN | 21.82 MPa | 60.44 MPa | 21.99 MPa |
| Physics-state GRU | 23.06 MPa | 66.00 MPa | 24.19 MPa |
| Physics-integrator NN | **0.0128 MPa** | **0.0237 MPa** | **0.0275 MPa** |

![Model comparison](artifacts/t2_multiaxial_ood_v2/model_protocol_comparison.png)

The physics-integrator NN predicts a bounded plastic-multiplier correction and
then applies differentiable J2/Chaboche consistency corrections while storing
plastic strain, PEEQ and two backstress tensors. It passed the mild and severe
structural gates with reaction relative-L2 errors of 1.65e-7 and 2.39e-7. The
severe case recorded one trust-region fallback at complete unloading.

This is a **white-box physics-fusion ceiling**: the embedded return-mapping
equations match the constitutive families that generated the synthetic data.
It demonstrates the benefit of architecture-level physics for this controlled
benchmark, but does not establish transfer to unknown materials or
misspecified physical models.

## Reproduction and evidence

- Dataset details: `data/t2_multiaxial_ood_v2/README.md`
- Model card: `models/t2_multiaxial_ood_v2/README.md`
- Quality report: `artifacts/t2_multiaxial_ood_v2/QUALITY_REPORT.md`
- Full metrics: `artifacts/t2_multiaxial_ood_v2/model_metrics.json`
- Structural gates: `artifacts/t2_multiaxial_ood_v2/structural_validation/`
- Manuscript roadmap: `docs/T2_MULTIAXIAL_RESEARCH_MEMO.md`
- Literature map: `docs/T2_MULTIAXIAL_LITERATURE.md`
- Commands: `T2_V2_REPRODUCE.md`
- DENIM data and evidence: `DENIM_START_HERE.md`
- DENIM reproduction: `DENIM_REPRODUCE.md`

The original multiaxial release used AgentFEM commit
`058faecc05aeda143d014fd229401003a9258bbb` (`0.3.7.dev0`). Later release
manifests record their own software revisions and hashes. All quantities use SI
units. Voigt order is `xx, yy, zz, xy, yz, xz`, with tensor shear.

## Scope

The releases are controlled synthetic, small-strain benchmarks for J2/Chaboche
families, material-parameter variation and one deliberately hidden
three-memory reference material. They are not a general metals database and
do not constitute experimental calibration, fatigue-life prediction, damage,
finite-strain plasticity or a production-certified learned material. Complete
internal states are privileged simulation labels, not quantities normally
available from standard material tests. Data are CC BY 4.0; code files are
covered by the included code license.


## Evidence protocol v3

See `article_evidence_v3/` for frozen ablation and leave-one-family-out split definitions.